  An initial-Neumann boundary value problem for a Keller--Segel system with
density-suppressed motility and source terms is considered. Infinite-time
blowup of the classical solution was previously observed for its source-free
version when dimension $N\geq2$. In this work, we prove that with any source
term involving a slightly super-linear degradation effect on the density, of a
growth order of $s\log s$ at most, the classical solution is uniformly-in-time
bounded when $N\leq3$, thus preventing the infinite-time explosion detected in
the source-free counter-part. The cornerstone of our proof lies in an improved
comparison argument and a construction of an entropy inequality.
