  We consider the classification problem for finite-dimensional irreducible
representations of the Yangians associated with the orthosymplectic Lie
superalgebras ${\frak{osp}}_{2n|2m}$ with $n\geqslant 2$. We give necessary
conditions for an irreducible highest weight representation to be
finite-dimensional. We conjecture that these conditions are also sufficient and
prove the conjecture for a class of representations with linear highest
weights. The arguments are based on a new type of odd reflections for the
Yangian associated with ${\frak{osp}}_{2|2}$. In the Appendix, we construct an
isomorphism between the Yangians associated with the Lie superalgebras
${\frak{osp}}_{2|2}$ and ${\frak{gl}}_{1|2}$.
