Profusely means ‘in a very large amount’, with that same pouring-out feeling. This adverb is the form many students will actually hear. He was sweating profusely (soaking wet). The wound bled profusely. She apologized profusely (she said sorry again and again). He thanked them profusely. Though actually, you cannot put it on every verb. Eat profusely and study profusely sound wrong. It wants verbs where something can pour or overflow: sweat, bleed, apologize, thank, praise. It is a little formal, but the sweat and apology uses are common even in news and everyday stories.