I assume that by non-degenerate you mean non-singular, and I assume that $k$ is a finite field.

The determinant of $M$, considered as a matrix over the polynomial ring $R=k[X_{11},\dots,X_{nn}]$, is a polynomial $f\in R$, and your problem is to determine whether $f$ defines the constant $0$ function over $k$.

There are several division-free algorithms for computation of determinant in any commutative ring using polynomially many ring operations. In principle, you can use such an algorithm to compute $f$, but this may result in a long expression, since $f$ may have exponentially many terms. It is better to combine this with a polynomial identity testing algorithm: since $f$ has degree $n$, the Schwartz–Zippel lemma tells you that $f(a_{11},\dots,a_{nn})\ne0$ for randomly chosen $a_{ij}\in k$ with probability at least $1-n/p$, as long as $f\ne0$ and $p>n$. Thus, if (say) $p>2n$, you don’t actually have to evaluate $f$, you have a simple probabilistic polynomial-time algorithm: choose random assignment to your variables, and test whether the resulting matrix is nonsingular.

In the special case where each $M_{ij}$ is either $0$ or $X_{ij}$, things are much simpler: it is easy to see that there is an assignment making the matrix nonsingular if and only if there is a permutation $\pi$ such that $M_{i\pi(i)}\ne0$ for each $i$, in other words, if and only if the bipartite graph whose adjacency matrix is defined from $M$ by replacing every $X_{ij}$ with $1$ has a perfect matching. There are various efficient algorithm for finding perfect matchings and/or checking their existence, see e.g. http://en.wikipedia.org/wiki/Perfect_matching#Algorithms_and_computational_complexity .