  \v{C}u\v{c}kovi\'{c} and Paudyal recently characterized the lattice of
invariant subspaces of the shift plus a complex Volterra operator on the
Hilbert space $H^2$ on the unit disk. Motivated by the idea of Ong, in this
paper, we give a complete characterization of the lattice of invariant
subspaces of the shift operator plus a positive integer multiple of the
Volterra operator on Hardy spaces $H^p$, which essentially extends their works
to the more general cases when $1\leq p<\infty$.
