Most generaly, does every continuous injective mapping $f:M\rightarrow N$ (these are smooth manifolds) have a smooth and regular injective approximation of arbitrary precision, given that $\dim(N)>\dim(M)$ and that $M$ is compact?
I don't really think that sharing my attempts at this is anywhere near helpful because they all look like overcomplicating misses, but I'll share something. Firstly, I only really considered the case where $N$ is an Euclidean space because it didn't seem to take away the essence of the question, and I assumed there is a measure on $M$ that agrees with standard measures on every cart (0 set in one measure is a 0 set in the other). Now, I was trying to develop some operators which would smooth out the function (much like the convolution does), but that will also retain the implicitness. If $\tilde f$ is the name of the to-be-built approximation, my idea was to define $\tilde f(x)$ to be the average of $f$ on $f^{-1}(B(f(x),\varepsilon))$ for some globally constant $\varepsilon$. It is possible to pick an $\varepsilon$ so that this $\tilde f$ is at least continuous (then I simply hoped it was smooth too), and I thought that for different $x$, the ball is shifted so the average shifts too, but it simply wasn't true (for $\dim(N)>1$). My second idea was only relevant in the case $\dim(M)=1$ (so, basically $M=\mathbb{S}^1$, a circle). I would take a smooth and regular approximation the is not necessarily injective, but I at least have that any two points on the circle that map to the same point are close on the circle and the preimage of any point is finite (because of the regularity and compactness). I would then take the preimage of some point and redefine the $\tilde f$ to be constant on the whole closed arc that contains the preimage. Then I move, say, clockwise until the next preimage and redefine it there too. At the end, my $\tilde f$ is piece-wise smooth or constant. Now I can play a little bit to make these constant corners smooth too. That was very informal, but I hope you get the main idea.
In general, you cannot make a smooth approximation preserving injectivity. As an example, consider $M$ which is a 7-dimensional exotic sphere, $N={\mathbb R}^8$. Since $M$ is homeomorphic to $S^7$, there exists a topological embedding $f: M\to N$. On the other hand, $M$ does not admit a smooth embedding in $N$, see my answer here. Thus, there are no regular smooth injective approximations of $f$.