I don't know specifically about homotopies, but the notion of a curved $A_\infty$-algebra is generally problematic. In the conventional setting of algebras over a field, it is just trivial in the following strong sense.

Let $A$ and $B$ be two curved $A_\infty$-algebras over a field $k$ with nonzero curvature elements $m_{0,A}\ne0\ne m_{0,B}$. This is a sufficient condition to trivialize nonunital curved $A_\infty$-algebras; in the (strictly) unital case, assume that $m_{0,A}$ and $m_{0,B}$ do not belong to the one-dimensional vector subspaces generated by the units of $A$ and $B$ (which could happen in the $\mathbb Z/2$-graded case).

Then any isomorphism of graded vector spaces $f\colon A\to B$ taking $m_{0,A}$ to $m_{0,B}$ (and preserving also the units, in the unital case) can be extended to an $A_\infty$-isomorphism $(f_0,f_1,f_2,\dotsc)\colon A\to B$ with $f_0=0$ and $f_1=f$. So there are precisely as many curved $A_\infty$-algebras with nonzero curvature, up to $A_\infty$-isomorphism, as there are graded vector spaces; and any curved $A_\infty$-algebra with a nonzero curvature is $A_\infty$-isomorphic to a curved $A_\infty$-algebra with $m_1=m_2=m_3=\dotsb=0$.

Similarly, any curved $A_\infty$-module over a (nonunital or strictly unital) curved $A_\infty$-algebra with a nonzero curvature element is contractible.

These results are mostly due to Kontsevich; I learned them from conversations with him while visiting IHES and subsequently recorded them in what is now AMS Memoir vol.212 #996, 2011, http://arxiv.org/abs/0905.2621, Remark 7.3.

It appears that if you want to have a nontrivial theory of curved $A_\infty$-algebras, you have to do it over, say, a local ring and require the curvature elements in your algebras to be divisible by the maximal ideal of the local ring. I am presently working on this; the writeup is available from my homepage.