Define for $m \in (0,1)$ fixed the sequence $$f_n(x) := \int_0^x |T_n\left((|y|-\frac{1}{n})^+ + \frac{1}{n}\right)\text{sign}(y)|^{m-1}$$ where we define $T_n(y) =y$ if $|y| \leq n$ and otherwise $T_n(y) = n$.
Does $f_n$ converge uniformly to $f(x) := \frac{1}{m}x|x|^{m-1}$? These functions are defined on $\mathbb{R}$.
I think it is enough to show that the integrand is uniformly convergent as a sequence, but I got no idea how to start.
Just use dominated convergence. $T_n(y)=y$ for $n\geq x$ and $y\in[0,x]$. You have $|(|y|-1/n)^++1/n|\leq |y|.$ Equivalently, you have the uniform convergence you desire because for $|y|<1/n$, your error is no more than $1/n$ and you have exact equality when $|y|>1/n$.
But is your question written correctly? There's a bunch of unnecessary issues in it, like $\mbox{sgn}(y)$ which has an absolute value around it.