Let $H\mathbb{Z}$ denote the spectrum for cohomology with coefficients in $\mathbb{Z}$, so your spectrum is $H\mathbb{Z}\wedge M_k(G)$.  Let $0\to\mathbb{Z}^I\stackrel{f}{\to}\mathbb{Z}^J\to G\to 0$ be a presentation of $G$.  Then we can explicitly construct a Moore space $M_k(G)$ as the cofiber of a map $\bigvee_I S^k\to \bigvee_J S^k$ whose induced map on homology is $f$.  In spectra, we then have a cofiber sequence $$\bigvee_I \Sigma^k H\mathbb{Z}\to\bigvee_J\Sigma^k H\mathbb{Z}\to H\mathbb{Z}\wedge M_k(G).$$

For any pointed finite CW-complex $X$, we then have a long exact sequence in cohomology of the form $$\dots\to\bigoplus_IH^{n+k}(X,\mathbb{Z})\to \bigoplus_J H^{n+k}(X,\mathbb{Z})\to H\mathbb{Z}\wedge M_k(G)^n(X)\to\dots$$

Now consider $X=S^0$.  The long exact sequence above computes that $H\mathbb{Z}\wedge M_k(G)^{-k}(S^0)=G$ and $H\mathbb{Z}\wedge M_k(G)^n(S^0)=0$ for $n\neq -k$.  Thus up to a degree shift, the cohomology theory $H\mathbb{Z}\wedge M_k(G)$ satisfies the Eilenberg-Steenrod dimension axiom, so it must coincide with $H^{*+k}(X,G)$.