  In this paper, we consider an initial-Neumann boundary value problem for a
parabolic-elliptic Keller-Segel system with signal-dependent motility and a
source term. Previous research has rigorously shown that the source-free
version of this system exhibits an infinite-time blowup phenomenon when
dimension $N \geq 2$. In the current work, when $N \leq 3$, we establish
uniform boundedness of global classical solutions with an additional source
term that involves slightly super-linear degradation effect on the density, of
a maximum growth order $s\log s$, unveiling a sufficient blowup suppression
mechanism. The motility function considered in our work takes a rather general
form compared with recent works \cite{FuJi2020, LyWa2023} which were restricted
to the monotone non-increasing case. The cornerstone of our proof lies in
deriving an upper bound for the second component of the system and an
entropy-like estimate, which are achieved through tricky comparison skills and
energy methods, respectively.
