This is from Discrete Mathematics and its Applications
This is the book means when mentions a list of common ways to express conditional statements
After going through the list, I immediately recognized "q is necessary for p". Translating to the "if p, then q" form, I got "If you want to get promoted, you must wash the boss's car.
Another interpretation of this that I got - from what I learned on Clarifying on how if p,q is logically equivalent to p only if q is
"You can get promoted only if you wash the boss's car". This means the only path to promotion is washing the boss's car. If you managed to get promoted(get there) without washing the car, the contract is broken.
Does all of that sound right?
Your statement is almost correct. In mathematics "only if" would mean "the only path to promotion includes washing the boss's car". Don't forget that "washing the boss' car" may not be enough to get "a path to promotion".