In this article we study the large $N$ asymptotics of complex moments of the absolute value of the characteristic polynomial of a $N\times N$ complex Ginibre random matrix with the characteristic polynomial evaluated at a point in the unit disk. More precisely, we calculate the large $N$ asymptotics of $E|det(GN-z)|\gamma$, where $GN$ is a $N\times N$ matrix whose entries are i.i.d and distributed as $N-1/2Z$, $Z$ being a standard complex Gaussian, $\Re (\gamma )>-2$, and . This expectation is proportional to the determinant of a complex moment matrix with a symbol which is supported in the whole complex plane and has a Fisher-Hartwig type of singularity: $det(\int Cwiw―j|w-z|\gamma e-N|w|2d2w)i,j=0N-1$. We study the asymptotics of this determinant using recent results due to Lee and Yang concerning the asymptotics of orthogonal polynomials with respect to the weight $|w-z|\gamma e-N|w|2d2w$ along with differential identities familiar from the study of asymptotics of Toeplitz and Hankel determinants with Fisher-Hartwig singularities. To our knowledge, even in the case of one singularity, the asymptotics of the determinant of such a moment matrix whose symbol has support in a two-dimensional set and a Fisher-Hartwig singularity, have been previously unknown.