Il n'y en avait pas beaucoup de présents.

English Translation

There were not many present.

There were not many present (people there). is not the same as
There were not many presents (gifts).
Which is a correct translation?

In this case, I’d tend to say that “présents” is the correct translation; if it was talking about gifts, the plural would be used, I think. Whereas with people it seems to be invariable (see oxford’s dictionary, second meaning: “There were 200 people present at the meeting.”).
Although the sentence seems a bit clunky. I think I would have said something like “Many were not present”