  We study the black hole spacetime structure of a model consisting of the
standard Maxwell theory and a $p$-power-Yang-Mills term. This non-linear
contribution introduces a non-Abelian charge into the global solution,
resulting in a modified structure of the standard Reissner-Nordstr\"{o}m black
hole. Specifically, we focus on the model with $p=1/2$, which gives rise to a
new type of modified Reissner-Nordstr\"{o}m black hole. For this class of black
holes, we compute the event horizon, the innermost stable circular orbit, and
the conditions to preserve the weak cosmic censorship conjecture. The latter
condition sets a well-established relation between the electric and the
Yang-Mills charges. As a first astrophysical implication, the accretion
properties of spherical steady flows are investigated in detail. Extensive
numerical examples of how the Yang-Mills charge affects the accretion process
of an isothermal fluid in comparison to the standard Reissner-Nordstr\"{o}m and
Schwarzschild black holes are displayed. Finally, analytical solutions in the
fully relativistic regime, along with numerical computations, of the mass
accretion rate for a polytropic fluid in terms of the electric and Yang-Mills
charges are obtained. As a main result, the mass accretion rate efficiency is
considerably improved, with respect to the standard Reissner-Nordstr\"{o}m and
Schwarzschild solutions, for negative values of the Yang-Mills charge.
