KL-divergence, total variation distance and weak convergence against unbounded functions

Question

Let $p_n,p$ be absolutely continuous probability measures on $\mathbb R^d$. Suppose they are close w.r.t. Kullback-Leibler divergence: $$ D_{KL}(p_n|p) \leq \frac{C}{n} \quad\forall\,n\geq1\,.$$ As a consequence they are also close in total variation: $$ \|p_n-p\|_{TV}^2\leq \frac{2C}{n}\quad\forall\,n\geq1\;.$$ What can we say about the difference $$ \Big|\int f\,dp_n-\int f\,d p\Big| $$ when $f:\mathbb R^d\to\mathbb R$ is continuous, $\sup_n\int|f|^l dp_n<\infty$ for every $l\geq1$, but $f$ not bounded ?