As defined, internal energy can be used to determine the heat transfer of a constant volume process, and enthalpy can be used to determine the heat transfer of a constant pressure process. Specific heats quantify how the energy in a substances varies with respect to temperature.
In particular, a constant volume specific heat indicates how the internal energy varies with respect to temperature (at a particular specific volume and temperature). A constant pressure specific heat indicates how the enthalpy varies with respect to temperature (at a particular pressure and temperature).
$$ c_v=(\frac{\partial u}{\partial T})_v~~~~~~c_p=(\frac{\partial h}{\partial T})_p $$
These values are summarized by lookup tables. However, the specific heats aren’t listed for every possible material, temperature, specific volume, and pressure combination because there are a number of assumptions we can reasonably make about how specific heats may be used.
It is reasonable apply an incompressible model in assuming that the specific volume of many solids and liquids cannot vary. In this case, internal energy varies only with respect to temperature. Further, enthalpy varies as a result of the variation in internal energy, and to a minor degree due to change in pressure with respect to temperature.
$$ v=c\Rarr u=u(T),~h=u(T)+p(T)v $$
Neglecting the variation in pressure with respect to temperature, we obtain that the variation in internal energy and enthalpy with respect temperature are identical, and the same specific heat may be used for both at a given temperature.
$$ u=u(T),~h=u(T)+pv\Rarr c=(\frac{\partial u}{\partial T})_v=(\frac{\partial h}{\partial T})_p $$
Assuming $c$ to vary negligibly with respect to temperature, we may finally calculate changes in internal energy and enthalpy for incompressible substances according to the following.
$$ \Delta u=\Delta h=c(\Delta T) $$
Since specific volume and specific internal energy vary negligibly with respect to pressure for liquids, the value of each for a saturated liquid at a given temperature may be used for a liquid at that temperature and any pressure. Enthalpy of a liquid may be calculated based on the sample principles, but accounting for increase in the pressure term, or further approximated in the same manner as specific volume and specific internal energy.
$$ v(T,p)\approx v_f(T)~~~~~u(T,p)\approx u_f(T) $$
$$ h(T,p)\approx h_f(T)+(p-p_{sat}(T))v_f(T)\approx h_f(T) $$