If I have a function $p:\tilde X\to X$ and a function $f:Y\to X$ , then a function $\tilde f:Y\to\tilde X$ such that $p\circ\tilde f=f$ is called a lift of $f$ with respect to $p$. So a lift is just a name for a solution of the equation $p\circ x=f.$ Is there a name for the solution of the equation $x\circ p=f$ (with the functions' domains and codomains set so that it makes sense)?
Its an extension.
The dual (with some relabelling) is: given a function $i : X \to \tilde X$ and a function $f : X \to Y$ we get a function $\tilde f : \tilde X \to Y$ such that $\tilde f \circ i = f$.
If $X \subseteq \tilde X$ and of $i$ is the inclusion map $X \hookrightarrow \tilde X$, this is just saying that $\tilde f$ is an extension of $f$ to $\tilde X$.
This notion of extension also makes sense when $i$ is an arbitrary injective function. It makes less sense when $i$ is not injective, but no less sense than the word lift when $p$ is not surjective.