There is a continuity problem with (a film)

The word is literally "discontinuous", as in a discontinuous mathematical function.

From M-W:

discontinuous : (1) not continuous (2) not continued; discrete <discontinuous features of terrain> b: lacking sequence or coherence 2: having one or more mathematical discontinuities —used of a variable or a function

If you mean continuity error in the sense of fiction, then the work is inconsistent or (stronger) self-contradictory.

inconsistent: having parts that disagree with each other : not in agreement with something

A continuity error is a mistake in the props or wardrobe of a scene, between two cuts.

Here's a typical explanation of a continuity error, from a Famous Film Critic, courtesy of Dan's reference above in the comments:

"Every movie has what are called continuity errors, those little inconsistencies obsessive film viewers collect, like a character’s hair parted one way and then, a cut later, parted the other."

• Reference: famous professional film critic in News Media

Note that this sentence clearly gives an example of a continuity error, i.e. a mistake in the props or wardrobe of a scene, between two cuts...

"like a character’s hair parted one way and then, a cut later, parted the other."

Note that - as you can see in this actual reference - a continuity error is: something that happens from one side to the other of a cut such as an object moving, etc.

As the writer says, it's a "little inconsistency" -- in a scene or cut -- and then the writer immediately - right after the comma - explains very explciitly that that is what is meant, a "little inconsistency" such as the hair going the wrong way a cut later

Noting carefully that

"a cut later"

is the actual language used there.

(Note that you could, if really wanting to be obtuse, peel away all of that sentence except one word ("inconsistencies") and see it as referring to something like, say, a "plot inconsistency" (example -- "But Darth was sterile, how could he gave children??" or the like). This has no connection to a "continuity error" which is when the hair goes the wrong way "a cut later".)

So, a "continuity error" is a problem (hair the wrong way, before and after a cut) in a scene or cut of a film.

There's no specific word for "a film with a lot of continuity errors". You'd just say "it has terribly continuity problems" or "it has a many continuity errors."

Note that analogously: a printed book can have "many typos" - there isn't really a word for "typo-laden."

Note that it is 100% normal in English that, with "two-word" descriptors, there is no way to describe that en masse, with only one word. You just have to use the two-word descriptor with "-laden" or "full of..." or "has many of...". Again this is completely normal and unsurprising in English.

A continuity error in a scene of a film ..

Note that just like say being a "grip," "dialog coach," "DOP," or whatever, "continuity person" is a common film occupation:

A humorous supposed quote, S Spielberg is known for leaving many continuity errors in the scenes he films. 'Fuck continuity!" is supposedly a humorous quote from him, but, I could not find a actual good reference for this.