$\newcommand\la\lambda$If $X_1,X_2,\dots$ are iid random variables (r.v.'s) each with the exponential distribution $E(\la)$ with parameter $\la$, then $\sum_1^n X_i\sim E(k,\la)$.
In turn, if $U_1,U_2,\dots$ are iid r.v.'s each with the uniform distribution over the interval $(0,1)$, then the r.v.'s $X_i:=-\dfrac1\la\,\ln U_i$ will be iid, each with the exponential distribution $E(\la)$. So, $$-\frac1\la\, \ln \prod_{i=1}^k U_i =\sum_{i=1}^k\Big(-\frac1\la\, \ln U_i\Big) =\sum_{i=1}^k X_i \sim E(k, \lambda),$$ as desired.
As for your modification of the Erlang distribution, it is quite unclear how you define it. In particular, what are $p$, $v$, and $\nu$? What is the set of possible values of $k$? Is the resulting function $f$ a density function? Is there a reference to such a modification of the Erlang distribution?