Okay, let's find the mistakes in these sentences!
1. Jenny are going to watch TV in the living room.
The mistake here is "are". "Jenny" is a singular noun (one person), so we need a singular verb. The correct verb to use with "Jenny" is "is".
Corrected Sentence: Jenny is going to watch TV in the living room.
2. Tom won't playing football on Friday.
The mistake here is "playing". After "won't" (which is a contraction of "will not"), we need the base form of the verb "play". Because of 'won't' we don't need the auxiliary verb 'be'.
Corrected Sentence: Tom won't play football on Friday.