  Real-space modulated Charge Density Waves (CDW) are an ubiquituous feature in
many families of superconductors. In particular, how CDW relates to
superconductivity is an active and open question that has recently gathered
much interest since CDWs have been discovered in many cuprates superconductors.
Here we show that disorder induced by proton irradiation is a full-fledged
tuning parameter that can bring essential information to answer this question
as it affects CDW and superconductivity with different and unequivocal
mechanisms. Specifically, in the model CDW superconductor Lu$_5$Ir$_4$Si$_{10}$
that develops a 1D CDW below 77\,K and s-wave superconductivity below 4\,K, we
show that disorder enhances the superconducting critical temperature
$T_\mathrm{c}$ and $H_\mathrm{c2}$ while it suppresses the CDW. Discussing how
disorder affects both superconductivity and the CDW, we make a compelling case
that superconductivity and CDW are competing for electronic density of states
at the Fermi level in Lu$_5$Ir$_4$Si$_{10}$, and we reconcile the results
obtained via the more common tuning parameters of pressure and doping. Owing to
its prototypical, 1D, Peierls type CDW and the s-wave, weak-coupling nature of
its superconductivity, this irradiation study of Lu$_5$Ir$_4$Si$_{10}$ provides
the basis to understand and extend such studies to the more complex cases of
density waves and superconductivity coexistence in heavy fermions, Fe-based or
cuprates superconductors.
