I found 8 of them and believe there is no more:

$$2+3^2=3+2^3$$ $$2+6^2=6+2^5$$ $$6+15^2=15+6^3$$ $$3+16^2=16+3^5$$ $$3+13^3=13+3^7$$ $$2+91^2=91+2^{13}$$ $$5+280^2=280+5^7$$ $$30+4930^2=4930+30^5$$

I call the solution a principal pipe of order 2. Any idea to attack this equation?

If $n=x=2$, it becomes $2+y^2=y+2^m$. It is Ramanujan and Nagell Equation, and there are only 3 solutions. If $n=2$ and $x \geq 3$, then we can use the similar way to prove that there is no other solution for $x < 10000$ with help of computer.

If $n=3$, and we assume $x$ and $y$ are prime, and $6\mid (m-1)$, then the solution is $3+13^3=13+3^7 = 2200$.

I believe there is no solution for $n>3$. I am particularly interested in solutions that both $x$ and $y$ are odd primes.