A person who writes plays is a playwright, not a playwrite, but the act of writing plays is usually spelled playwriting. The wright in the compound noun playwright is a little-used word referring to one who constructs or repairs something. It also appears in shipwright, which refers to a person who builds ships.
Playwrighting is a rare and old variant, registered in some dictionaries, but it usually gives way to playwriting in edited texts.
Of course, playright is a misspelling.
In 1962 which yet-to-be-famous playwright was given a six-month jail term for defacing library books? [The Independent]
It’s always distracting when you can hear the cogs of self-conscious cleverness grinding behind a playwright’s conceit. [Los Angeles Times]
It’s one thing for a playwright to have a reading of a play, but what if you’re a seven-headed playwright whose work constantly evolves? [Calgary Herald]