Representations of the super Yangians of types $A$ and $C$

We classify the finite-dimensional irreducible representations of the super Yangian associated with the orthosymplectic Lie superalgebra ${\frak osp}_{2|2n}$. The classification is given in terms of the highest weights and Drinfeld polynomials. We also include an $R$-matrix construction of the polynomial evaluation modules over the Yangian associated with the Lie superalgebra ${\frak gl}_{m|n}$, as an appendix. This is a super-version of the well-known construction for the ${\frak gl}_n$ Yangian and it relies on the Schur--Sergeev duality.