어깨에 기대다 (eo-ggae-e gi-dae-da) to lie/lean on (someone’s) shoulder
Firstly, do not mix up 기대다 (to lean) with 기대하다 (to look forward/expect/anticipate).
Another thing is to take note that the particle attached to 어깨 (shoulder) or any other nouns like 벽 (wall) or 팔꿈치 (elbow) or even another person is ~에 and not 와/과 or 을/를.
This is one of the few (or many) exceptions where we can use English to explain.
You wouldn’t lean with someone/something or lean someone/something but rather, lean on someone/something.
Therefore, it is important to remember that we must use ~에.
너무 피곤해 보여서 내 어깨에 기대 줄래?
You look really tired, shall I lend you my shoulder to lie on?
선생님은 그 학생과 얘기할 때마다 벽에 기대고 바닥만 바라봐요.
Whenever our teacher talks to the student, he would just lean against the wall and look at the floor.