That small band you are seeing are short oligos, either RNA or DNA, that are present in your mini prep. Your gel is not going to resolve small fragments well, so it probably represents a number of sizes.
Longer RNase treatment might remove the band if they are digested RNA (which would be my best guess).
The plasmid is contaminated, or degraded (exposed to heat, light, chemicals). Use high quality reagents and follow the purification protocol carefully.