Hope for explanations I'm a bit worried, I'm resident of Kuwait with my maiden name and recently I have received new passport with my marriage name. Can I apply from Kuwait for U.K.visa with my new passport and no evidence inside it about Kuwait residence
UK embassy/consulate will almost certainly require you to provide proof of residence in Kuwait to order to apply for a visa from its territory.
Having a valid visa in an expired passport is not an issue. Simply include old passport and a copy of marriage certificate (and any papers proving the name change if there are any) with your visa application.