"Weakly" Woodin cardinals

(As I pointed out in a comment) yes, partial Woodinness is common in arguments in inner model theory. Accordingly, you obtain determinacy results addressing specific pointclasses (typically, well beyond projective). To illustrate this, let me "randomly" highlight two examples:

• See here for $\Sigma^1_2$-Woodin cardinals and, more generally, the notion of a cardinal $\delta$ being Woodin with respect to a family $\frak A\subseteq \mathcal P(\delta)$.
• See here for $\Gamma$-Woodin cardinals (and coarse mice) for $\Gamma$ a good pointclass.

There are also more recent examples, of course.

There is a useful notion of "weak Woodinness" according to which every uncountable regular cardinal is weakly Woodin :) See https://ivv5hpp.uni-muenster.de/u/rds/fabiana_ralf.pdf