Let $p\in(0,1]$, $q\in(1,\infty)$, $\alpha\in[n(1-1/q),\infty)$ and $w_1,\,w_2\in A_1$. The author proves that the norms in weighted Herz-type Hardy spaces $HK^{\alpha,\,p}_q(w_1,\,w_2)$ and $HK^{\alpha,\,p}_q(w_1,\,w_2)$ can be achieved by finite central atomic decompositions in some dense subspaces of them. As an application, the author proves that if $T$ is a sublinear operator and maps all central $(\alpha,\, q,\,s;\,w_1,\,w_2)_0$-atoms (resp. central $(\alpha,\, q,\,s;\,w_1,\,w_2)$-atoms of restrict type) into uniformly bounded elements of certain quasi-Banach space $\cal B$ for certain nonnegative integer $s$ no less than the integer part of $\alpha-n(1-1/q)$, then $T$ uniquely extends to a bounded operator from $H\dot K^{\alpha,\,p}_q(w_1,\,w_2)$ (resp. $HK^{\alpha,\, p}_q(w_1,\,w_2)$) to $\cal B$.