Learn Turkish: Onun kullanması gereken pompa bozuk. - The pump she must use is broken.

This is the classic formula for a relative clause of necessity. It literally means "that is necessary for her to use". It combines a verbal noun with a possessive suffix (kullan-ma-sı, meaning "her using") with gereken (meaning "that is necessary"). Placed right before the noun pompa, it creates "the pump she must use."

Can I drop the word Onun and just say Kullanması gereken pompa bozuk?

Yes, absolutely. Because the word kullanması already contains the third-person possessive suffix (-sı), it is already clear that we mean "his" or "her" using. Including Onun at the beginning simply adds emphasis, contrasting her required pump with someone else's.

I know "the pump she uses" is kullandığı pompa. Why don't we use the -diği suffix here, like kullanması gerektiği pompa?

This is a common trap! In kullandığı pompa, she is the one doing the using, so we use the object relative clause suffix (-diği). But in this necessity structure, you are describing the pump as "the pump that is necessary (for her to use)." Because the state of being necessary describes the pump itself, gerek takes the subject relative suffix -en.

Why use bozuk for "broken" instead of kırık?