Settlement date, deposit size and clean conditions are leverage. A cover letter that reveals your ceiling is not.
Price is only one lever. A settlement date that suits the seller, a sensible deposit, and conditions with short timeframes can win against a higher price with messy terms. Deadlines in your offer put pressure on the seller to decide.
A letter to the seller can help, but only if it does not give away how much you want the house or how far you would go.
A free 15-minute call with a property lawyer who only works with first home buyers.