Gavi, Nest Warden
Legendary Creature — Human Shaman
You may pay 
rather than pay the cycling cost of the first card you cycle each turn.
Whenever you draw your second card each turn, create a 2/2 red and white Dinosaur Cat creature token.
A mother's love transcends species.
Illustrated by
Commander 2020#7MythicEnglishLegality
Rulings
If a cycling cost includes 
and you pay an alternative activation cost of instead, X is 0.
Gavi’s first ability considers the entire turn. If you cycle a card then have Gavi come under your control, you can’t pay to cycle your second card.
The triggered ability can trigger only once each turn. It doesn’t matter Gavi was on the battlefield when the first card was drawn. If it’s not on the battlefield when the second card is drawn, the ability can’t trigger at all that turn. It won’t trigger when the third or fourth card is drawn.
If a spell or ability causes you to put cards into your hand without specifically using the word “draw,” it’s not a card drawn.