What complexity class does this variation of traveling salesman problem belong to?
I'm assuming that you're thinking of the Euclidean traveling salesman problem, where $c$ and $v$ are vectors in $\mathbb{Z}^{2n}$, with two coordinates for each city.  Let $minTSP(c)$ denote the length of the minimum traveling salesman tour for the cities with coordinates $c$.  Then your problem asks whether
$$
minTSP(c + v) \ge min TSP(c) + x.
$$
But then the special case $c=0$ is equivalent to asking whether $minTSP(v) \ge x$, which is coNP-hard.