Unmaximized Hamiltonian Inclusion (UHI) type conditions are derived for optimal control problems with mixed constraints. For such problems different necessary conditions of optimality, including UHI type conditions, have previously been proved assuming that the Jacobian of the mixed constraint functional with respect to the control variable have full rank. Here we show that such requirement can be replaced by possibly weaker conditions, namely, by Mangasarian Fromowitz type conditions. Not ably we consider problems in which the dynamics is nonsooth. The Unmaximized Hamiltonian Inclusion type conditions we present here are written as a weak version of nonsmoothmaximum principle stated in terms of a joint Clarke subdifferential. They are of interest since, incontrast to more traditional non- smooth maximum principles, they give sufficiency for normal linear convex problems.