Yes, you are right, sorry (I guess you are talking about one of the two components of the convective term).
The enthalpy equation does not have a pressure gradient term. In order to see that, you can take the (total) energy equation as a starting point, and then applying the definition h = E - e - p/rho - v^2/2 you will get an equation without a pressure gradient term (e is referred to the internal energy)
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