？
Would it be possible to extend my stay?
Literal
Stay-extension [object-を] request-can-do [question-か]?
延泊 ('extending a hotel stay') — 延 'extend' + 泊 'overnight stay' — is standard hotel vocabulary. お願いできますか is an indirect request form: 'would I be able to ask you?' — softer than お願いします because it frames the request as a question about possibility.