Writs are usually legal documents issued by the courts, mandating some sort of behavior from a person. In "The Rocking Horse Winner," the family is always short on money, and yet they live a lifestyle like they were rich. The mother, instead of being frugal and wise with her money, spends it on tutors and fancy things for the house. As a result, there is never enough money, and they are always falling short.
Paul, in a conversation with his uncle about the situation declares, "You know people send mother writs, don't you uncle?" His uncle does indeed know that. This means, most likely, that Paul's mother owes a lot of people money and is not paying them back. So, they have to go through the courts, and demand that she pay them. It's like if I owed you $3,000 for damaging your car, but never paid. You could take me to court and sue me, and the courts would require, legally, that I pay you. That is what was happening to Paul's mother; she was getting writs from people that she owed money.
It's a rather sad situation, and one that is the foundation for Paul's obsession with his rocking horse and finding winning horses. Unfortunately, as is the case with greed, even when he wins, it isn't enough, and the price that it demands is tragic. I hope that helped; good luck!