  We obtain new bounds for short sums of isotypic trace functions associated to
some sheaf modulo prime $p$ of bounded conductor, twisted by the Mobius
function and also by the generalised divisor function. These trace functions
include Kloosterman sums and several other classical number theoretic objects.
Our bounds are nontrivial for intervals of length at least
$p^{1/2+\varepsilon}$ with an arbitrary fixed $\varepsilon >0$, which is
shorter than the length at least $p^{3/4+\varepsilon}$ in the case of the
Mobius function and at least $p^{2/3+\varepsilon}$ in the case of the divisor
function required in recent results of {\'E}.~Fouvry, E.~Kowalski and P.~Michel
(2014) and E.~Kowalski, P. ~Michel and W.~Sawin (2018), respectively.
