I paid part of the rent after an eviction notice, does that help my case?

It might do more than you'd expect. Accepting rent after the notice already went out can function as an implied waiver, meaning the landlord may have given up his right to rely on that notice, for that period, which pulls the rug out from under the whole basis for the case.

Partial payments get complicated fast depending on exact timing and what was communicated when the payment happened, so this isn't a clean, universal rule. Hold onto every receipt, bank record, and text exchanged around the payment, since the specifics of when and how it happened are what actually decide whether this argument works for you.