The first quoted sentence is grammatically incorrect, because it combines two independent clauses without proper punctuation or a coordinating conjunction. Here's a breakdown of the sentence:
"What do you usually do during that period?" - This is an
interrogative sentence asking about someone's activities during a
specific time frame. It is grammatically correct on its own.
"Since I have no idea what I’ll do." - This is a dependent clause
that provides a reason for asking the question in the first
sentence. However, it cannot stand alone as a sentence, and it
should be connected to the main clause with a comma, not a period.
To make the sentence grammatically correct, you can combine the two clauses using a comma:
"What do you usually do during that period, since I have no idea what I’ll do?"
Thus, your second sentence correct.