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Suppose $\mathfrak{C}$ is an abelian category which does not have enough projectives and we're interested in computing the right derived functors of some covariant functor $F$.

If however, every object $N$ admits an $F$-acyclic resolution, then can we still compute the homology of $N$ using some $F$-acyclic resolution, even if there are no projective resolutions of that object?

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Yup, check out the theory of $F$-projectives and you'll note that taking your categroy $\mathscr{P}$ of F-projectives to be exactly those acyclics, will be sufficient for your purouses :)

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