For two square matrices $P$ and $Q$ over skew fields, the explicit formulas for the Drazin inverse of $P+Q$ are given in the cases of (i) $PQ^2=0$, $P^2QP=0$, $(QP)^2=0$; (ii) $P^2QP=0$, $P^3Q=0$, $Q^2=0$, which extend the results in [M.F. Martínez--Serrano et al. On the Drazin inverse of block matrices and generalized Schur complement, Appl. Math. Comput.] and [C. Deng et al., New additive results for the generalized Drazin inverse, J. Math. Anal. Appl.]. By using these formulas, the representations for the Drazin inverse of $2\times2$ block matrices over skew fields are obtained, which also extend some existing results.