Yes, hydrogen can absorb neutrons and emit the released energy as a gamma-ray photon. Your value is a bit too large, however. The Table of Isotopes (Firestone 1996) lists the neutron separation energy of deuterium (which is the same process in the other direction) as 2224.57 keV.
As to the practical feasibility for fusion, where would you get the neutrons from? Assuming you somehow got hold of some, you could let them react with hydrogen, e.g. in a water basin. This happens in fission reactors, but only as a side effect. The energy released by neutron absorption is only a small fraction of the total. One 235U fission releases about 200 MeV, the few extra neutrons get you a few MeV only.
Free neutrons are produced in D+T fusion, but if you can do that, your argument is kind of unnecessary, since you have fusion anyway. So the question is, can you get free neutrons for less energy input than you get back out from their absorption minus conversion losses? I don’t think so.