  The concept of a covering system was first introduced by Erd\H{o}s in 1950.
Since their introduction, a lot of the research regarding covering systems has
focused on the existence of covering systems with certain restrictions on the
moduli. Arguably, the most famous open question regarding covering systems is
the odd covering problem. In this paper, we explore a variation of the odd
covering problem, allowing a single odd prime to appear as a modulus in the
covering more than once, while all other moduli are distinct, odd, and greater
than $1$. We also consider this variation while further requiring the moduli of
the covering system to be square-free.
