Let $X$ denote a magma and suppose $x \in X$. Then clearly, if there exists a magma $Y$ and an injective homomorphism $f : X \rightarrow Y$ such that $f(x)$ has a left-inverse, then it follows that $x$ is left-cancellative.
Does the converse hold?
Meaning, is it true that if $x \in X$ is left-cancellative, then there exists a magma $Y$ and an injective homomorphism $f : X \rightarrow Y$ such that $f(x)$ has a left-inverse?
My definitions are as follows.
"$x$ is left-cancellative" =
$$\forall a,b : x*a = x*b \rightarrow a=b$$
"$x$ has a left-inverse" =
$$\exists y \forall a : y*(x*a)=a.$$
You can simply pick $Y = X \cup \{x^{-1}\}$ and find a way to extend $*$ to $Y$ :
Define $x^{-1} * (x*a) = a$ forall $a \in X$ (this is possible since $x \mapsto x * a$ is injective). For $x * x^{-1}$, pick any result in $Y \setminus \{ x * a, a \in X\}$, so you can define next $x^{-1} * (x * x^{-1}) = x^{-1}$. For the remaining operations involving $x^{-1}$, you don't have any constraint so anything you pick works.