If M, A are two differential graded complexes over a commutative ring R with the following data, $$(M,d_M) \overset{\Delta}{\longrightarrow} (A,d_A)$$ $$(A,d_A) \overset{f}{\longrightarrow} (M,d_M)$$ $$A \overset{\phi}{\longrightarrow} A$$ with the following conditions, $$f \Delta=1_{M}, \Delta f=1_{A}-(d_{A} \phi+\phi d_{A}) , \phi \Delta=0, f \phi=0$$. Such a data is called Strong Deformation Retract Data(SDR).
In this setup, suppose t is a perturbation of $$d_{A}$$ i.e ($$d_A+t$$ )is a new differential in A which amounts to saying t satisfies $$[d_A,t]=0 , t^{2}=0$$. Then Homotopy Transfer lemma allows you to define a new differential on $$(M,\Delta_{\infty})$$ so that $$(M,\Delta_{\infty}) , (A,d_A+t)$$ are SDR pair.
I wanted to know if there is any Homotopy Transfer Lemma without the hypothesis $$f\Delta=1$$. What if there is another map $$\psi: M \rightarrow M$$ satisfying $$f\Delta=1_{M} -(d_{M}\psi+\psi d_{M})$$? Can we still expect such a lemma? If yes, can we write down an explicit formula for the new differential as before? Please point me to the right reference. Thank you!