I will present results establishing cancellation in short sums of arithmetic functions (in particular the von Mangoldt and divisor functions) twisted by polynomial exponential phases, or more general nilsequence phases. These results imply the Gowers uniformity of suitably normalized versions of these functions in intervals of length $X^{c}$ around $X$ for suitable values of $c$ (depending on the function and on whether one considers all or almost all short sums). I will also discuss applications to averaged forms of the Hardy-Littlewood conjecture. This is based on joint works with Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao and Terence Tao.