Coupling a spinor field to a preexisting scalar field?

So I’m not a physicist, but I’m thinking about a mathematical problem where I think physical insight might be useful.

We’re working on a Riemannian manifold $(M,g)$ (positive definite metric) with a distinguished smooth function $f$. The metric and the smooth function are related by a tensor equation $$Rc(g)+nabla^2f=displaystylefrac{1}{2}g$$ (The second summand is the Hessian wrt the Levi-Civita connection.)

I want to study spinor fields $psi$ which solve some kind of Dirac equation but somehow involve this function $f$. (Is it appropriate to say I want to “couple” $psi$ and $f$?) I thought perhaps physicists had thought about such things and may have insights.

So, my question: Are there natural equations (from a physics POV) to write down for $psi$ that involve $f$?

I’m aware of the Yukawa interaction (thanks Google), which is a Lagrangian you can write down for undetermined scalar and matter fields, but in this case the scalar field is fixed ahead of time, so I don’t know how that figures in.

Any thoughts at all are appreciated.