The separation order was likely made so that the wife had proof that she and the children had been deserted and she would be able to apply for maintenance from her husband. If the husband didn't pay up he could be taken to court and fined or sent to prison. If the husband couldn't be found, his wife could apply for poor relief. She might also have approached local charities or church for financial or practical help. All these bodies needed to know that her request was genuine.
Which country? English family law was that a father had legal rights over the children of his marriage. There were several changes to this law between late 19th and mid 20th centuries so you might like to check the situation in 1902.
Grounds for divorce were restrictive, even more so for a wife than husband. It was also expensive, uncommon and scandalous. Divorce cases were probably reported in press, so the scandal became public.
