Well, you can always take the "Free $2$-category generated by $C$ (as a category) and $2$-arrows of the form $f \Rightarrow F(f)$". So that the $2$-arrow will be "all the thing you can form by composite $2$-arrows of the form $f \to F(f)$. And any $2$-category looking something like what you are after will have a $2$-functor from this free object.

However, this free object does not appear to have a nice description - and the discusion below strongly suggest to me that no such $2$-category will have a nice description :

For example, because in a $2$-category you need to be able to (vertically) compose $2$-arrows you'll also have $2$-arrows of the form $f \Rightarrow F^2(f)$ and $f \Rightarrow F^3(f)$ etc... but then horizontal composition of $2$-arrows means that if an arrow $h$ can be written as a composite you'll also have $2$-arrows of the form

$$ h = f \circ g \Rightarrow F(f) \circ F^2(g) $$

Which you can't simply express this as a $h \Rightarrow F^k(h)$, so it follows that the existence of a $2$-arrow $h_1 \Rightarrow h_2$ is going to at least depends in a relatively complicated way on how $h_1$ and $h_2$ can be factored as composition.

And in facts things are even worst than this: because if $F(f) \circ F^2(g)$ can be itself written as a composite $u \circ v$ in a different way, then you'll have a further map

$$ f \circ g \Rightarrow F(f) \circ F^2(g) = u \circ v \Rightarrow F^4(u) \circ v $$

(for exemple, the $F^4$ is completely random) which cannot even be understand simply in terms of how $h = f \circ g$ factors only, it also involve factorization of $F(f) \circ F^2(g)$

Well, you can always take the "Free $2$-category generated by $C$ (as a category) and $2$-arrows of the form $f \Rightarrow F(f)$". So that the $2$-arrow will be "all the thing you can form by composing $2$-arrows of the form $f \to F(f)$". And any $2$-category looking something like what you are after will have a $2$-functor from this free object.

However, this free object does not appear to have a nice description - and the discusion below strongly suggest to me that no such $2$-category will have a nice description :

For example, because in a $2$-category you need to be able to (vertically) compose $2$-arrows you'll also have $2$-arrows of the form $f \Rightarrow F^2(f)$ and $f \Rightarrow F^3(f)$ etc... but then horizontal composition of $2$-arrows means that if an arrow $h$ can be written as a composite you'll also have $2$-arrows of the form

$$ h = f \circ g \Rightarrow F(f) \circ F^2(g) $$

Which you can't simply express this as a $h \Rightarrow F^k(h)$, so it follows that the existence of a $2$-arrow $h_1 \Rightarrow h_2$ is going to at least depends in a relatively complicated way on how $h_1$ and $h_2$ can be factored as composition.

And in facts things are even worst than this: because if $F(f) \circ F^2(g)$ can be itself written as a composite $u \circ v$ in a different way, then you'll have a further map

$$ f \circ g \Rightarrow F(f) \circ F^2(g) = u \circ v \Rightarrow F^4(u) \circ v $$

(for exemple, the $F^4$ is completely random) which cannot even be understand simply in terms of how $h = f \circ g$ factors only, it also involve factorization of $F(f) \circ F^2(g)$