Let $H(p) = \sum_i p_i\log\frac{1}{p_i}$ be the entropy of $p$ and $KL(p, q) = \sum_i p_i\log\frac{p_i}{q_i}$ be the KL divergence between $p$ and $q$. Does it hold that $H(p) \le H(q) + KL(p, q)$?

If this is not true, can we bound $H(p)$ using $H(q)$ and $KL(p, q)$ in certain form?