What is the object of these prepositions? [closed]

As you had guessed, and as Lambie confirmed in the comments, the object of both prepositions in your sentence is the following coordination of noun phrases (NPs):

the manufacture, processing, packing, or holding of a drug.

For what it's worth, I agree that the writing in the text you cited is simply atrocious.

The commas in your sentence are arguably necessary (or at least strongly favored). However, they are not an essential part of this type of construction, as the examples below show (e.g. see [21] ii and esp. [22] i). Rather, the decision of whether to put the commas or not is guided by the usual considerations that govern comma placement in general (which is a fairly complex subject in its own right).

General description of this type of construction

The construction is traditionally called right node raising (RNR). It is famous for presenting significant challenges to linguists. For example, the analysis that gave it its name is no longer widely accepted.

CGEL calls it delayed right constituent coordination (p. 1343):

In this construction the constituent which in basic coordination would appear as the rightmost element of the first coordinate is held back until after the final coordinate:

[21] i She knew of m͟y͟ ͟o͟t͟h͟e͟r͟ ͟w͟o͟r͟k͟ but never mentioned it. [basic coordination]
       ii She knew of but never mentioned m͟y͟ ͟o͟t͟h͟e͟r͟ ͟w͟o͟r͟k͟.    [delayed right constituent]

In general, the effect is to heighten the contrast between the coordinates by removing from them material that would be the same in each. But the construction is appreciably more difficult to process than basic coordination, both for the addressee, who has to hold the first coordinate in mind until the sense is completed at the end, and for the speaker, who has to plan ahead to ensure that each coordinate ends in a way that syntactically allows completion by the delayed element—as knew of and never mentioned both allow completion by an NP complement.[Examples are found where this condition is not satisfied. One case is illustrated in ^?I always have and always will v͟a͟l͟u͟e͟ ͟h͟e͟r͟ ͟a͟d͟v͟i͟c͟e͟, where the plain form value is an admissible continuation of will but not of have: compare basic I have always v͟a͟l͟u͟e͟d͟ her advice and always will v͟a͟l͟u͟e͟ it. Another involves coordination of comparisons of equality and inequality: ^?It's as good or better t͟h͟a͟n͟ ͟t͟h͟e͟ ͟o͟f͟f͟i͟c͟i͟a͟l͟ ͟v͟e͟r͟s͟i͟o͟n͟, where as good takes a complement headed by as not than. Such examples are not fully grammatical and would generally be avoided in monitored speech and writing; the second can be corrected to It's as good as or better than t͟h͟e͟ ͟o͟f͟f͟i͟c͟i͟a͟l͟ ͟v͟e͟r͟s͟i͟o͟n͟.] Characteristically, there is a prosodic break after the final coordinate, signalling that the element that follows relates to the whole coordination, not just to the final coordinate.

Examples of RNR

Here are some further paradigmatic examples of that sort of construction. It has five or so major types of use, explained below (from CGEL pp. 1343-1344, slightly paraphrased). Your example is just like that in [22] i; note the absence of commas in that one, however.

(a) Heads taking different complementation.

[22] i I'm i͟n͟t͟e͟r͟e͟s͟t͟e͟d͟ ͟i͟n͟ but r͟a͟t͟h͟e͟r͟ ͟a͟p͟p͟r͟e͟h͟e͟n͟s͟i͟v͟e͟ ͟a͟b͟o͟u͟t͟ their new proposal.
       ii He o͟u͟g͟h͟t͟ ͟t͟o͟, but p͟r͟o͟b͟a͟b͟l͟y͟ ͟w͟o͟n͟'͟t͟, make a public apology.

In [i] the heads interested and apprehensive take different prepositions, in and about. In [ii] ought and will take different kinds of infinitival complement, with to appearing only with ought.

(b) Contrasts of time or modality, The same verb in each coordinate but with differences expressing such concepts as time or modality:

[23] i She w͟a͟s͟ ͟t͟h͟e͟n͟, i͟s͟ ͟n͟o͟w͟, and a͟l͟w͟a͟y͟s͟ ͟w͟i͟l͟l͟ ͟b͟e͟, devoted to the cause of peace.
       ii They r͟e͟g͟a͟r͟d͟e͟d͟ ͟h͟i͟m͟, or a͟p͟p͟e͟a͟r͟e͟d͟ ͟t͟o͟ ͟r͟e͟g͟a͟r͟d͟ ͟h͟i͟m͟, as a complete liability.

(c) Pairing of lexically simple and lexically complex coordinates. The first coordinates here are simple verbs, while the final ones are complex expressions. [ii] illustrates the fairly common case where one or more of the coordinates in this construction is a verbal idiom:

[24] i He had either t͟e͟l͟e͟p͟h͟o͟n͟e͟d͟ or w͟r͟i͟t͟t͟e͟n͟ ͟a͟ ͟l͟e͟t͟t͟e͟r͟ ͟t͟o͟ his son's boss.
       ii You should w͟e͟l͟c͟o͟m͟e͟, not t͟a͟k͟e͟ ͟o͟f͟f͟e͟n͟c͟e͟ ͟a͟t͟, the suggestions they make.
      iii He was a͟c͟c͟u͟s͟e͟d͟ but f͟o͟u͟n͟d͟ ͟n͟o͟t͟ ͟g͟u͟i͟l͟t͟y͟ of stalking a woman for seven years.

(d) Contrasting sequences of pre-head dependents in noun phrase (NP) structure.

[25] i They have f͟i͟v͟e͟ ͟n͟e͟w͟ and t͟w͟o͟ ͟s͟e͟c͟o͟n͟d͟-͟h͟a͟n͟d͟ copies of his novel.
       ii Neither t͟h͟e͟ ͟A͟m͟e͟r͟i͟c͟a͟n͟ nor t͟h͟e͟ ͟R͟u͟s͟s͟i͟a͟n͟ people want war.

In [i] both dependents in the coordinated sequences contrast, whereas in [ii] only the second does. The repetition of the in [ii] is motivated by neither which could not occur between the and American, but such repetition is not restricted to cases where it is syntactically or semantically required: He was comparing t͟h͟e͟ ͟A͟m͟e͟r͟i͟c͟a͟n͟ and t͟h͟e͟ ͟R͟u͟s͟s͟i͟a͟n͟ versions.

(e) Contrasting pairs of subject and verb.

[26] i K͟i͟m͟ ͟m͟a͟y͟ ͟a͟c͟c͟e͟p͟t͟, but P͟a͟t͟ ͟w͟i͟l͟l͟ ͟c͟e͟r͟t͟a͟i͟n͟l͟y͟ ͟r͟e͟j͟e͟c͟t͟, the management's new proposal.

Not limited to coordination

However, even CGEL's name 'delayed right constituent coordination' is a bit of a misnomer, because RNR is not limited to coordination ('coordination' = joining by one of and, but, or, ...]. (There is also another reason, to be discussed in the final section of this answer.) For example, in the following sentences, the delayed right constituent occurs in a subordinative construction rather than in a coordination (CGEL, p. 1344):

[27] i I͟ ͟e͟n͟j͟o͟y͟e͟d͟, although e͟v͟e͟r͟y͟o͟n͟e͟ ͟e͟l͟s͟e͟ ͟s͟e͟e͟m͟e͟d͟ ͟t͟o͟ ͟f͟i͟n͟d͟ ͟f͟a͟u͟l͟t͟ ͟w͟i͟t͟h͟, her new novel.
      ii T͟h͟o͟s͟e͟ ͟w͟h͟o͟ ͟v͟o͟t͟e͟d͟ ͟a͟g͟a͟i͟n͟s͟t͟ far outnumbered t͟h͟o͟s͟e͟ ͟w͟h͟o͟ ͟v͟o͟t͟e͟d͟ ͟f͟o͟r͟ my father's motion.

So it seems to me that a better name might be delayed right constituent construction. But my impression is that 'RNR' is, in fact, a pretty standard name for this.

Coordination as evidence of constituenthood, in the light of RNR

This is best explained by an example. Consider the sentence (CGEL, p. 1348)

[36] Sue found the key.

Linguists say that found the key is here a single constituent. Which word sequences are syntactical constituents is a basic question in syntactical analysis. (If found the key is indeed a constituent, further arguments will show that it is of a type that linguists call a verb phrase (VP), whose head is the verb found.)

We would like to defend this 'VP analysis' against the competing one which says that there is no such constituent: that all we have is a verb, found, and an NP, the key, which however do not together form any larger syntactical entity.

One major piece of evidence in favor of the VP analysis is that found the key can be readily coordinated:

[37] Sue f͟o͟u͟n͟d͟ ͟t͟h͟e͟ ͟k͟e͟y͟ and u͟n͟l͟o͟c͟k͟e͟d͟ ͟t͟h͟e͟ ͟d͟o͟o͟r͟.

The basis of that argument is the following apparently plausible principle: 'if a sequence X can be coordinated with a sequence Y to form a coordination X and Y, then X and Y are constituents' (CGEL, p. 1348).

This can be challenged on several grounds, one of which (rather famously) involves RNR: namely, it is also possible for coordination to group found with Sue: M͟a͟x͟ ͟l͟o͟s͟t͟ but S͟u͟e͟ ͟f͟o͟u͟n͟d͟ the key. 'If coordination can group found with either the key or Sue, the argument would go, then it can't provide evidence for a constituent grouping of found with just one of them, the key' (CGEL, p. 1349).

Nevertheless, following a lengthy discussion on pp. 1348-1350 (in which objections other than those from RNR are considered as well), CGEL concludes that the argument in favor of the VP analysis—the argument based on [37]—does ultimately succeed. The reason is that cases like RNR, where coordinates are not constituents, can be shown to be inapplicable to [37]. The final lesson, according to CGEL, is this: 'if a sequence X can be coordinated, then the simplest account will be one where it is a constituent entering into basic coordination, and we will adopt some other, more complex, analysis only if there are independent reasons for doing so.'

Cases where the delayed element is not a constituent

Note that in all of the examples above, the delayed group of words was a constituent. However, it is possible to find examples where this is not so, i.e. where the delayed group of words is a mere sequence of words (CGEL, p. 1343):

It had to be ascertained whether the managers had suitable people to put forward for possible appointment from persons [registered with, or applying to, t͟h͟e͟m͟ ͟f͟o͟r͟ ͟e͟m͟p͟l͟o͟y͟m͟e͟n͟t͟].

This is yet another reason why the name 'delayed right constituent coordination' is a misnomer: not only the construction need not be a coordination, but also the words that are delayed need not form a constituent at all (although they do so in most cases). So, apparently, we would need to call it something like 'the construction of a delayed right constituent or sequence'... which is not a good name, to say the least. So perhaps it is indeed better to stick with tradition and call it RNR.