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Let's extract a clear question (about spectra in general) from your question, and then answer it. Let $E$ be any spectrum. There is the degree $p$ map $p:S\to S$ from the sphere spectrum to itself. ...

If your category has enough projectives, choose a projective resolution of $M$, and write $\Omega^nM$ for the $n$th kernel in this resolution, say $$0 \to \Omega^nM \to P_{n-1} \to \cdots \to P_1\to ...

Elements of $\operatorname{Ext}^i(M,N)$ for $i \geq 1$ can be represented by Yoneda extensions: exact sequences $$E = \big(0 \to N \to Z_{i-1} \to \ldots \to Z_0 \to M \to 0\big)$$ modulo the ...