Do photon emissions increase the gravity of the Sun by compressing it
In principle the momentum of the outgoing light would imply some force exerting some inward pressure on the sun, but in practice the resulting compression would be negligible. However, even if that were a significant effect, it wouldn't increase the sun's gravity because the gravitational force created by a spherical body depends only on the mass of the body and the distance from its center. The only reason that we think of a compressed dense planet as having stronger gravity than a less dense less compressed planet is that we're closer to the center when we stand on the surface of the denser one... But you could compress the sun down to the size of a black hole, or blow it up until it reached the orbit of Mercury, and as long its mass stayed the same its gravitational pull on the earth would be unchanged.
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