If a person wants to do an act of worship and detects insincerity in himself, should he go ahead and carry on that act of worship and struggle with one’s sincerity?

Yes it’s better to struggle with the act than to give up because Shaytan comes to us and puts doubts in us so that we find elements of insincerity in every act, because to the degree one is sincere, one is rewarded. So, if we are sincere 50 % then we will get a reward of 50%. Therefore, don’t give up and abandon acts when one has doubts.

As it is mentioned in the hadith: Prophet Muhammad (ﷺ) said: “A slave may pray and have nothing recorded for it except a tenth of it, or a ninth, or an eighth, or a seventh, or a sixth, or a fifth, or a quarter, or a third, or a half.” [1]

Reference:

[1] Reported by Imaam Ahmad; Saheeh al-Jaami’, 1626