How would you describe Paul’s relationship with his mother?

Unfortunately, this relationship is quite one-sided. Paul desperately seeks the love and affections of his mother, but she doesn't reciprocate these feelings.
In the beginning of the story, the mother's lack of maternal instincts is made clear:

She had bonny children, yet she felt they had been thrust upon her, and she could not love them. They looked at her coldly, as if they were finding fault with her . . . When her children were present, she always felt the centre of her heart go hard.

Paul's mother makes great pretenses about her love of her children, trying with tremendous and unnatural effort to seem gentle with them and act concerned for them. However, she knows in her heart that she feels no love for anyone, not even her own children. Thus, she does not love Paul.
In contrast, Paul picks up on the anxious feelings of his mother and senses that she longs for a better life. He is willing to do whatever it takes to provide for his mother. Through his efforts, he presents his mother with 5,000 pounds to alleviate some of her tension (worth at least 200,000 pounds today, adjusted for inflation). This does not win his mother's affection. He continues to crave it, conveying this through his comment, "Didn't you have anything nice in the post for your birthday, mother?" She responds that it was moderately nice with a tone that is "cold and absent."
In the end, Paul dies trying to make his mother happy, without ever knowing the warmth of a mother's love.