Is the money a supplicant takes from his mother lawful, and is he sinful if he contents himself with reading and learning and abandons earning a livelihood, knowing that his mother is in need of someone to care for her?

There is no harm in you caring for your mother, pursuing knowledge, and accepting what she gives you. However, you must strive to secure a profession or a job to support yourself in the future, and balance between work and caring for your mother.

"And whoever fears Allah – He will make for him a way out And will provide for him from where he does not expect. And whoever relies upon Allah – then He is sufficient for him. Indeed, Allah will accomplish His purpose. Allah has already set for everything a [decreed] extent." (At-Talaq/2-3)