I am trying to understand whether some non trivial explicit estimate is known about the following
$$\sum\_{n \leq x} \mu(n) \chi(n)$$
as a function of $x$.


Here $\chi$ is a Dirichlet character modulo $q$ and $q > (\log x)^2$ (for example). 



Sum of Möbius function multiplied by a character
Estimating this sum is much the same as estimating the error term in the prime number theorem for arithmetic progressions. See Exercises 7-8 in Section 11.3 of Montgomery-Vaughan: Multiplicative number theory I (Cambridge University Press, 2006). (Your sum is denoted by $M(X,\chi)$ in this book, as introduced by (11.39) there.) 


In particular, there is a constant $c>0$ such that for any $A>0$ we have
$$ \sum\_{n \leq X} \mu(n) \chi(n)\ll\_A x\exp(-c\sqrt{\log x})$$
as long as $q\leq(\log x)^A$ .


