Translation:Do you have a handkerchief?
Handkerchief o arimasu ka to me sounds like 'Is there a handkerchief?' Maybe it can also be used when asking 'do you have' as in 'do you possess' but am not sure.
__ o motteimasu ka = do you own _? but I think it would be weird in English to say 'own' in some situations, so 'have' is used in translations
For example, in okane o motteimasu ka, it's weird to say 'do you own/possess money?
When translating from English and the sentence uses 'have,' I check if the context is looking for existence/having vs owning/possession to choose between arimasu/imasu and motteimasu.
Do you have kids? = The question seems to ask about existence of kids. Besides, kids aren't owned, so unless we're dealing with child slavery, I'll stick with 'imasu'
Do you have a car? = Here it's about possession, so I'll use motteimasu.
At least, that's how I see it.