I'm a layman as well, but I think photons can effectively lose energy in various ways, one of them being the well-known redshift effect. And one of the main causes of redshifting is photons climbing out of a gravitational well (this is called a gravitational redshift).

However, whether that translates to a loss in momentum is a bit more fuzzy and I really can't tell whether it does or doesn't. Although I'm far from being a physicist, so hopefully someone more knowledgeable chimes in to enlighten us...

But I'm curious: what is your reasoning for asking whether photons have a mechanism to lose momentum as a consequence of them affecting the curvature of spacetime? It's not at all obvious to me the relationship between these two concepts.