  In this paper, we study $(\mathcal F,\mathcal F')_{p}$-harmonic maps between
foliated Riemannian manifolds $(M,g,\mathcal F)$ and $(M',g',\mathcal F')$. A
$(\mathcal F,\mathcal F')_{p}$-harmonic map $\phi:(M,g,\mathcal F)\to (M',
g',\mathcal F')$ is a critical point of the transversal $p$-energy functional
$E_{B,p}$. Trivially, $(\mathcal F,\mathcal F')_2$-harmonic map is $(\mathcal
F,\mathcal F')$-harmonic map, which is a critical point of $E_B$. There is
another definition of a harmonic map on foliated Riemannian manifolds, called
transversally harmonic map, which is a solution of the Euler-Largrange equation
$\tau_b(\phi)=0$. Two definitions are not equivalent, but if $\mathcal F$ is
minimal, then two definitons are equivalent. Firstly, we give the first and
second variational formulas for $(\mathcal F,\mathcal F')_{p}$-harmonic maps.
Next, we investigate the generalized Weitzenb\"ock type formula and the
Liouville type theorem for $(\mathcal F,\mathcal F')_{p}$-harmonic map.
