When you do something wearily, you do it with great exhaustion, like finishing a big project for school late at night when you're so tired that your eyes can barely stay open.
Coming from the Old English word werig, meaning "exhausted," wearily shows the effort and struggle to keep going when energy is low, painting a clear picture of extreme tiredness. A traveler might walk wearily at the end of a long journey, each step feeling heavier than the last. Similarly, a student might complete their final exam wearily, their energy drained after hours of concentration.