$\newcommand\R{\mathbb R}$Let $f$ be a radial pdf on $\R^n$, so that 
$$f(x)=g(|x|)\tag{1}\label{1}$$
for some function $g\colon[0,\infty)\to\R$ and all $x\in\R^n$, where $|x|:=|x|_2$. Then your desired respresentation 
$$f(x)=\int_0^\infty h(w) \cdot 1_{w \mathbb B^n} \ dw\tag{2}\label{2}$$ with $h\ge0$ can be rewritten as 
$$g(|x|)=\int_0^\infty dw\,h(w)\,1(|x|\le w)$$
for all $x\in\R^n$ and then further as 
$$g(u)=\int_u^\infty dw\,h(w)$$
for all real $u\ge0$, with $h\ge0$. 

So, your desired representation \eqref{2} for $f$ as in \eqref{1} holds if and only if $g$ is a nonincreasing absolutely continuous function such that $g(u)\to0$ as $u\to\infty$. Moreover, then $h=-g'$ almost everywhere.