On the ECNL website, competition rule 2.12 outlines what a Discovery Player is. I can find no reference to any limitation on training days for a Discovery Player in the rules, nor am I aware of any. I am confident there is none.
However, ENCL rules dictate that a player who generates a "Travel Distance Concern", meaning they are unlikely to be able to regularly attend trainings, can be designated a Discovery Player by the Commissioner, and must take one of the team's Discovery Player slots. Why? I expect this insures the ECNL team is drawing and developing players in primarily their own market, and not recruiting half of their roster from distant areas, resulting in a game day roster very different than their training day roster.
The USSDA Developmental Player program (not Discovery) is a very different beast, with different rules and goals, and allows clubs to try out new players in up to 6 games (but no more) before adding them to the full time roster.