I applied for a job. (Sense of 'for'=asked for something in writing)
I applied for my friend. (Sense of 'for'=on behalf of)
Are those sentence correct?
Can I use 'for' in this way, because already a fixed preposition is there?
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Yes, you can "apply for someone" meaning "on their behalf". It just implies that the thing you're applying for is elided.
— Congratulations, honey! You got the waiter job!
— Huh? I didn't even apply for it.
— Oh, I applied for you. I figured it was about time you started working during the summers.
By the way, the sense you supply for "for" in the first case doesn't make sense. It has nothing to do with writing. If you want to supply a synonym for that "for", it would be "to request".