  The objective of this manuscript is to investigate the traversable wormhole
solutions in the background of the $f(R, \phi)$ theory of gravity, where $R$ is
the Ricci scalar and $\phi$ is the scalar potential respectively. For this
reason, we use the Karmarkar criterion for traversable static wormhole geometry
to create a wormhole shape function. The suggested shape function creates
wormhole geometry that links two asymptotically flat spacetime regions and
meets the necessary requirements. The embedding diagram in three-dimensional
Euclidean space is also discussed in order to demonstrate the wormhole
configurations. For our current analysis, we choose the suitable values of free
parameters for $f(R, \phi)$ gravity models to discuss the wormhole geometry. It
can be observed that our proposed shape function provides the wormhole
solutions with less amount of exotic matter. It can be noticed that energy
conditions especially null energy conditions are violated for all considered
models. The violation of energy conditions indicates the existence of exotic
matter and wormhole geometry. It is concluded that the shape function acquired
through the Karmarkar technique yields validated wormhole configurations with
even less exotic matter correlating to the chosen $f(R, \phi)$ gravity models.
