Will such an algorithm be feasible, will it be considered a poly time algorithm?
As the length of the input is $p$ and the worst case of the algorithm is $O(p\log(p))$, the algorithm is polynomial as the worst case is $O(p^2)$ and $p^2$ is polynomial.

For the second case, there is not any change in time complexity as constant values are multiplied or divided in the polynomial or logarithmic case. To scrutinize more, $p \frac{\log(p/5)}{100} = p \frac{(\log(p) - \log(5))}{100} = \frac{p\log(p) - p\log(5)}{100} = \Theta(p\log(p))$.