Let $V$ be a linear space. Consider a contractive linear map $M:V\mapsto V$, $$ \|Mv\|\leq \|v\| \quad \text{for all vectors } v\in V. $$ Now, for some fixed vector $c\in V$, the question is to sort out whether or not $$ \|Mv+c\|\leq \|v+c\| \quad \text{for all vectors } v\in V $$ is also true.
UPDATE: Sorry, it is stated, is trivially not true. Suppose $M v= v/2$, take $v=-c$, then $\|M (-c)+c\|=\|c\|/2>0=\|v+c\|$.
Consider instead the case that $M$ does not change the sum of components of $v$, $\sum_i (Mv)_i=\sum_i v_i$ for all $v\in V$?
In general no. Take $Mv = \frac 12 v$. Fix $c \in V$ (nonzero, since otherwise the inequality holds trivially) and let $v = -c$. Then $$\|Mv + c\| = \frac 12 \|c\|$$ but $$\|v+c\| = 0.$$