The answer is no. In fact, $d_H(A,C_n(L_n))$ can be however large or even infinite for all $n$.
Indeed, e.g. suppose that $X=[-1,a)$ for some $a\in(\frac12,\infty]$, $\ell(x)=u(x)=-x$ for $x\in[-1,0]$, $\ell(x)=u(x)=0$ for $x\in[0,a)$, $p_n=\frac1n$, and $L_n=\frac1{2n}$. Then $A\supseteq[0,a)$ and $C_n(L_n)=[-\frac3{2n},-\frac1{2n}]$, so that $$d_H(A,C_n(L_n))\ge d\Big(a-\frac1{2n},-\frac1{2n}\Big)=a$$ if $a<\infty$, and $d_H(A,C_n(L_n))=\infty$ if $a=\infty$.