、。
Could you please take care of my dog tomorrow?
Literal
Tomorrow, my [possessive-の] dog [possessive-の] care [object-を] do humbly-receive give-favor-not [polite-ませんか].
世話をする = 'to take care of' (literally 'do care'). The request uses ~していただけませんか — even more polite than ~ていただけますか, with the negative adding a softening 'won't you.' A polite favor request to a neighbor or friend, maybe a pet-sitting arrangement.