Do you use the expression "heartfelt hug" or " hearty hug", or something like any expression to hug heartily?

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Not really. To describe a nice hug, I’d say “a warm hug” or “a loving hug.”

For example: She gave her son a warm hug before he left.

Hope this helped!

Answer

Hmmm.   Not really.  There’s nothing wrong with those adjectives with hug, but I don’t believe it’s a very common usage.  Sounds odd somehow.  I wish I could be more specific.  If you’re hugging someone it seems redundant to add those modifiers.