Can monics always be "made left-invertible" by adding arrows?

No. Consider the special case where $C$ is a monoid in which every morphism is monic and epic, i.e. a cancellative monoid. If such a $D$ existed, then the subcategory of $D$ generated by $C$ and the inverses of morphisms of $C$ would be a group in which $C$ embeds, since every morphism gets an inverse on both sides. However, by a theorem of Malcev there exist cancellative monoids that cannot be embedded in a group.

It may be worth pointing out that the answer to the weaker question of "Is there a faithful functor $F:C\to C'$ sending monos to split monos" is positive. It's usually a better idea to try to only "improve" either limit-type or colimit-type constructions in a category-doing both at once gets you into trouble as in Eric's answer.

Indeed, we can just take the composite of the Yoneda embedding $C\to [C^{\mathrm{op}},\mathrm{Set}]$ with functor of precomposition along the inclusion of the set of objects (viewed as a discrete category) $i:\mathrm{ob}(C)\to C^{\mathrm{op}}$. The resulting functor $i^*:[C^{\mathrm{op}},\mathrm{Set}]\to [\mathrm {ob} C,\mathrm{Set}]$ is faithful, but not full; both the Yoneda embedding and $i^*$ preserve and reflect monomorphisms, and in $[\mathrm {ob} C,\mathrm{Set}]$, which is isomorphism to a product of copies of the category of sets. EDIT: Composing with one more functor the (faithful) disjoint basepoint functor $\mathrm{Set}\to \mathrm{Set}_*$ into the category of pointed sets, gives a faithful functor from $C$ into a product of copies of $\mathrm{Set}_*$, in which every monomorphism splits.

This is just a partial answer but is way too long to fit in a comment so here goes.

If a monomorphism has a left inverse then it's called a split monomorphism, similarly for epimorphisms if they have a right inverse then they are called split epimorphisms. Call a category in which every mono and epi splits a split category. Then we can define a "universal" split category for $C$ with a universal property. Call $F:C \rightarrow D$ a left split functor if monos in $C$ get taken to split monos in $D$ and a right split functor if epis in $C$ get taken to split epis in $D$, we call $F$ split if it's both left and right split.

We say that $C_{\text{split}}$ is a split completion for $C$ if there exists an $i:C \rightarrow C_{\text{split}}$ which is split and for any split functor $F$ there exists a unique functor $C_{\text{split}} \rightarrow D$ which makes the following triangle commute.

If $C_{\text{split}}$ exists, it is unique up to isomorphism of categories.

This definition is inspired by the localization of categories with respect to some collection of morphisms. If $(C,W)$ is a category with weak equivalences then we can define the localization $C[W^{-1}]$ defined as the category we obtain by formally adding morphisms to $C$ to make all morphisms in $W$ into isomorphisms. $C[W^{-1}]$ is known as the localization of $C$ at $W$. It's worth noting that $C[W^{-1}]$ doesn't always exist. It does always exist however if we permit hom classes instead of hom sets.

We can make an example computation of the split completion for the category with two objects $A,B$ and only one nontrivial morphism which is $f:A \rightarrow B$ to make sure we get what we expect. The category $C_{\text{split}}$ has two objects $A,B$ and two nontrivial morphisms $f$ and $f^{-1}$ where $f$ is an isomorphism $A \rightarrow B$ with $f^{-1}$ as its inverse.

I leave it up to you to fill in the proof but keep in mind that $f$ is both a monomorphism and an epimorphism.

If $C$ is small then $C_{\text{split}}$ will always exist, we can construct it as the category freely generated by the arrows in $C$ and for each mono $m$ we add $_l m$ going in the opposite direction to $m$ and for each epi $e$ we add $e_r$ going in the opposite direction to $e$ and then quotient the hom sets w.r.t the relations in $C$ and $_l m \circ m = \text{id}$ and $e \circ e_r = \text{id}$. You can read about the freely generated category on a quiver (directed multigraph) here.

This is a proof similar to the one showing that $C[W^{-1}]$ exists if $C$ is small.

Hope you found this at least somewhat helpful!

Edit: Note that we don't require that the functor $C \rightarrow C_{\text{split}}$ be faitful, thus my answer doesn't contradict Eric Wofseys.