What step does Paul take to ease his mother’s anxiety over the family’s debt?

The problem is that Paul will never truly be able to ease his mother's anxiety over debt, no matter how many times he rides that rocking-horse. As it becomes painfully apparent quite early on, no amount of money is ever enough for Hester. She's so riven with financial insecurity that she can never be satisfied with Paul's winnings.
She senses, perhaps, that this rare stroke of good luck won't last and so the only way to guard against future financial disaster is through the acquisition of more and more money. Hester curses herself over the bad luck she's encountered throughout her life, so it's not entirely unreasonable for her to think that past is prologue and that her bad luck will soon return. But bad luck does return, though not quite in the form that Hester could've imagined it. Tragically, her greed and insecurity lead to Paul's death, as he suffers a fatal aneurysm while riding himself into a frenzy on the rocking-horse.

Paul realizes that by riding his rocking horse, he can predict which horse will win at the races. He knows his mother feels the family is short of money.
His Uncle Oscar finds out Paul knows what horses are racing from Bassett, the young gardener. Then, by riding his rocking horse, the name of the winner comes to Paul.
When Paul has accumulated five thousand pounds, quite a lot of money in those days, he has his uncle arrange to give it to his mother, through the family lawyer, as a present from a relative.
Paul thinks this will satisfy his mother, but she quickly spends it. It only feeds her endless appetite for more. Therefore, Paul bets enough money on Malabar, who is racing in the Derby, to win his mother eighty thousand pounds. However, he dies soon after.