Degree of a determinant

Assuming that by degree you mean degree of the numerator minus the degree of the denominator, this is a trivial calculation. The determinant is $$frac{1}{(x^{p^{n+1}}-x)(x^{p^{n+2}}-x^p)}-frac{1}{(x^{p^{n+2}}-x)(x^{p^{n+1}}-x^p)}.$$ Multiplying the denominators of these fractions to combine them gives a denominator of degree $2(p^{n+1}+p^{n+2})$. In the combined numerator, the $x^{p^{n+1}+p^{n+2}}$ terms will cancel, leaving the highest degree term in the numerator as $x^{p^{n+2}+p}$. So the degree of the numerator minus the degree of the denominator is $$(p^{n+2}+p)-2(p^{n+1}+p^{n+2})=p-(p+2)p^{n+1}.$$