What Grammatical name and Grammatical function is given to the part of sentence in parenthesis

Solution 1:

Assume the original sentence is:

Traders who used to come early to the market claimed he had probably had some place he sleeps

Then the function of the relative clause (marked in bold) is to identify the subset of traders being talked about; namely, those who used to come early - as opposed to those who used to come later.

You can take your pick as to how you name such a relative clause:

The Cambridge Grammar Of English (p566) calls it a defining relative clause.

The Cambridge Grammar Of The English Language (p1034) calls it an integrated relative clause.

Oxford Modern English Grammar (p198) calls it a restrictive relative clause.

Assuming, alternatively, that the original sentence sets off the relative clause with commas or parentheses, then the function of the clause is to give extra information that applies to all traders - as opposed to other visitors to the market:

Traders, who used to come early to the market, claimed he had probably had some place he sleeps.

The three grammars listed above name such 'additional information' clauses non-defining, supplementary and non-restrictive, respectively.

A further point, a non-restrictive (etc.) relative clause, i.e. one that is set off by commas in writing, will normally be said a lower pitch than its restrictive equivalent.