Let $X$ be a smooth projective complex variety and $D$ a divisor with simple normal crossings on $X$, with irreducible components $D_i$. If $D_1$ is one of these (smooth) irreducible components, then one has an exact sequence of sheaves $$ 0 \to \Omega^p_X(\log D)(-D_1) \to \Omega^p_X(\log(D-D_1)) \to \Omega^p_{D_1}(\log (D-D_1)_{|D_1}) \to 0 $$ (This is explained e.g. in section 2 of the book by Esnault and Viehweg).

I'm wondering what happens if you remove several irreducible components at the same time. That is, let $H$ a reduced divisor with irreducible components some of the $D_i$. Is there a similar exact sequence relating $\Omega^p_X(\log D)(-H)$ and $\Omega^p_X(\log(D-H))$? Of course the problem is that $H$ is not smooth anymore, so I don't know how to make sense of the last term... Thanks for your help!