The Matching Cut problem is to decide if the vertex set of a connected graph can be partitioned into two non-empty sets $B$ and $R$ such that the edges between $B$ and $R$ form a matching, that is, every vertex in $B$ has at most one neighbour in $R$, and vice versa. If for some integer $d\geq 1$, we allow every neighbour in $B$ to have at most $d$ neighbours in $R$, and vice versa, we obtain the more general problem $d$-Cut. It is known that $d$-Cut is NP-complete for every $d\geq 1$. However, for claw-free graphs, it is only known that $d$-Cut is polynomial-time solvable for $d=1$ and NP-complete for $d\geq 3$. We resolve the missing case $d=2$ by proving NP-completeness. This follows from our more general study, in which we also bound the maximum degree. That is, we prove that for every $d\geq 2$, $d$-Cut, restricted to claw-free graphs of maximum degree $p$, is constant-time solvable if $p\leq 2d+1$ and NP-complete if $p\geq 2d+3$. Moreover, in the former case, we can find a $d$-cut in linear time. We also show how our positive results for claw-free graphs can be generalized to $S_{1^t,l}$-free graphs where $S_{1^t,l}$ is the graph obtained from a star on $t+2$ vertices by subdividing one of its edges exactly $l$ times.