No, "at a slow pace" does not work here. The verb "to run early" means to be ahead of schedule. But that's not what *Corinna mane currit". It means, I think, "Corinna gets up early in the day to go for a run". I would probably render that as "Corinna runs in the early morning" or "Corinna runs early in the day/in the morning".
Yes, here definitely something like "in the morning" is missing. Mane denotes primarily the first part of the day or as an adverb (early) morning. I would translate Corinna runs early, ie. before scheduled time, like "Corrina mature/matura currit." or maybe "Corinna tempestive/tempestiva currit."