Please bring a cup of tea to me.
FYI: お and ご are prefixes for polite expressions. But we often use お茶 and ご飯 (rice or meal in general) without the nuance of politeness.
The sentence is a polite imperative form, so you definitely need お with 茶. But we also frequently say 私はお茶を飲みました = I drank some tea. The doer is yourself, so you don’t need to attach お to respect for yourself at all. But we do attach it.
You can consider お茶 and ご飯 as being evolved in a different way from the original function, and they are just a set phrase in the modern era. This is just my personal thought but I think もったいない spirit (= don’t waste precious resources) is always woven into the bottom of our heart. Food and beverage are brought from the great nature. In short, it is derived from our belief in animism.