Transferred Chern classes in Morava $K$-theory

Let $\eta$ be a complex $n$-plane bundle over the total space of a cyclic covering of prime index $p$. We show that for $k\in \{1,2,...,np\} \setminus \{p,2p,...,np \}$ the $k$-th Chern class of the transferred bundle differs from a certain transferred class $\omega_k$ of $\eta$ by a polynomial in the Chern classes $c_p,...,c_{np}$ of the transferred bundle. The polynomials are defined by the formal group law and certain equalities in $K(s)^*B(Z/p \times U(n))$.