The author studies the \(F\)-energy functional \[ E_F(\varphi )= \int_M F\left(\tfrac{1}{2} \text{ trace }_{G_\theta} (\pi_H \varphi^*h) \right) \theta \wedge (d\theta )^n \] for any smooth map \(\varphi : (M,\theta)\rightarrow(N,h)\), where \(M\) is a strictly pseudoconvex CR manifold with a contact form \(\theta\), and \(N\) is a Riemannian manifold with a metric \(h\). She gives the first variation formula of the \(F\)-energy and some characterizations of \(F\)-pseudoharmonic maps, i.e., the critical maps of the \(F\)-energy functional.