Generalization of connections

Question

Let $A$ be an endomorphism of $1$-forms over a manifold $M$. I define the following action of the gauge group over $A$:

$$g^* A= f. g^{-1} dg + g^{-1}Ag$$

where $f\in {\cal C}^{\infty}(M)$ is a fixed smooth function, and $g$ is in the gauge group. We have:

$$(g_1 g_2)^* A= g_2^*( g_1^* A)$$

So we can define a generalization of connections as $f$ is not supposed to be inversible.

$$\nabla = f.d +A$$

$$\nabla (gs)= f(dg \otimes s)+ g\nabla (s)$$

What is the space of such generalization of connections?