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お茶でも飲んで

have some tea

→ 御茶
phrase Rare #45,574
have some tea でも makes the suggestion sound casual and noncommittal, like “some tea or something.” The て-form can connect this action to what follows.
JA: お茶でも飲んで、少し休みましょう。
EN: Have some tea and let’s take a short break.