Actually photons interact with spacetime by way of gravity.

First, their trajectories follow spacetime curvature. So bends close to stars, say, according to distant observers but not according to themselves, they travel in “straight lines”.

Second, they have energy if not mass, and that is also something that generates gravity changes. They bend spacetime right back at it, even if very, very little.

I don’t know if mass and time interacts in any deeper sense, but they do in the sense of general relativity as partly described in the article. Energy, whether as mass or not, tells spacetime how to curve. And spacetime curvature tells energy where to go. When energy goes wherever it goes, individual systems experience relativistic effects such as own local times. (E.g. ‘time dilation’ when compared to other observers at lower speeds or gravities.)