  We show existence of small solitary and periodic traveling-wave solutions in
Sobolev spaces ${\mathrm{H}^s}$, ${ s > 0 }$, to a class of nonlinear,
dispersive evolution equations of the form \begin{equation*} u_t + \left(Lu+
n(u)\right)_x = 0, \end{equation*} where the dispersion ${L}$ is a
negative-order Fourier multiplier whose symbol is of KdV type at low
frequencies and has integrable Fourier inverse ${ K }$ and the nonlinearity
${n}$ is inhomogeneous, locally Lipschitz and of superlinear growth at the
origin. This generalises earlier work by Ehrnstr\"om, Groves & Wahl\'en on a
class of equations which includes Whitham's model equation for surface gravity
water waves featuring the exact linear dispersion relation. Tools involve
constrained variational methods, Lions' concentration-compactness principle, a
strong fractional chain rule for composition operators of low relative
regularity, and a cut-off argument for ${n}$ which enables us to go below the
typical ${s > \frac{1}{2}}$ regime. We also demonstrate that these solutions
are either waves of elevation or waves of depression when ${ K }$ is
nonnegative, and provide a nonexistence result when ${ n }$ is too strong.
