@joro Exactly. That is what I have done.

As usual, the question concerning rational solutions is easier. In my paper: Rational points on certain del Pezzo surfaces of degree one, GLASGOW MATH J vol. 50 (2008), 557-564, I proved that for any $a, b$ the Diophantine equation $x^2+ay^5-z^6=b$ has infinitely many rational solutions.

Some solutions are omitted. If $n$ is odd and $x=0$ then we need to take $v=\pm 1$. However, then we get $y=0$ from our parametrization. This is not a real problem because we can parametrize $y^2=z^{n-1}$ easily. The same situation occurs in the case $n$ even and $v=0$. In both cases the reason for this is simple: the maps constructed are rational and not defined everywhere.

I believe that you may be interested in my recent paper {\it On the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$}, J NUMBER THEORY vol. 208 (2020), 418-440 (joint with H. R. Gallegos-Ruiz, N. Katspis and Sz. Tengely), where the problem of near binomial coefficients is investigated.

The smallest $c$ such that the Diophantine equation $3^xy+x=c$ has three positive solutions is $c=38180350917190281105854137945200663220016794$. The solutions correspond to $x=1, 4, 85$.

A related (and more difficult) equation: $a^3b^4+c^3d^4=e^3f^4$ is studied by A. Nitaj in the paper {\it On a conjecture of Erdos on 3-powerful numbers}, Bulletin of the London Mathematical Society 27 (1995), pp. 317-318.

Thomas, I think that the simplest way is to find me, say, on ResearchGate and I will send you a copy of the paper.