Hi Jessica, ~느라고 is used to explain why you couldn't to something. 그림을 그리느라고 놀지 못했다 = you were to busy drawing/painting, so you had no time to play. You wish you had more time to play, but you haven't. ~(으)려고 is used when you tell what the purpose of the action was. 그림을 그리려고 안 놀았다 = you didn't play because you wanted to draw/paint. You could choose. 뛰어가느라고 전화를 못 받았다. (Because I was running, I couldn't answer the call.) But you wouldn't say 뛰어가려고 전화를 못 받았다.

Hello Jessica, '~ 하느라(고)' means 'because of doing' 그리고 '~하려고' means 'in order to do' I carefully reckon you need to understand '안' and '못.' '안' means 'you intend not to do' and '못' means 'you cannot do' For example, 1). 그림을 그리느라고 놀지 못 했다 (o) = I could not hang out / play because I had to draw drawing (o) 그림을 그리려고 놀지 못 했다 (x) = I could not hang out / play in order to draw drawing (x) 그림을 그리려고 안 놀았다 / 놀지 않았다 (o) = I did not hang out / play in order to draw drawing (o) 2). 숙제를 하느라고 텔레비전을 안 봤다 (x) = I did not watch TV because I had to do homework (x) 숙제를 하느라고 텔레비전을 못 봤다 (o) = I could not watch TV because I had to do homework (o) 숙제를 하려고 텔레비전을 안 봤다 (o) = I did not watch TV in order to do homework (o) I hope this will help you. Please inform me of if you do not understand or there is wrong info, thank you :)

@Emma.K 감사합니다! 그건은 제가 알고 싶은 것이였어요. 그런데 1번이 왜 어색한지 더 자세히 설명할 수 있나요?

1번의 하려고는 문장이 어색한듯 해요

~느라고는 ~때문에 못했어 니까 더 부정적의견이 강해요. 핑계대는거라고 생각하면 이해하기 편할것 같아요 하려고는 목적이 강하구요 앞을 선택하여서 뒤를 하지않았다가 되요.