The person who told you that you had to remove the offsets has no idea what she is talking about. She assumed that since it was not blooming, the reason was the new plants.

You plant looks like it could be Mammillaria senilis. (There are a bunch of white spined Mammillarias. Without bloom, with this the only photo, it is taking a leap assigning any name to it). Most Mamms bloom in the spring. They also like nights about 20 degrees F cooler than the days, and most Mamms like it quite cool at night in the winter. A lot of them are frost tolerant if kept dry.

Also, a plant with a thick coat of spines needs really really strong light. A rule of thumb is that the stronger the spination, the stronger the light you need.

Overall, I would say that if you keep your plant dry in the winter in cool tamps in strong sun, you will be rewarded with rings of red/white flowers in the spring. If you keep it in a warm place, you will need to water it more, and it likely will lessen the bloom. If you keep it in a dark place, you might not get any flowers at all.

Thanks Ditzmoindonc. I'm glad you have said there is no need to remove the offshoots because I like them on the plant. I shall take on board your advice regarding watering needs. Once again, many thanks
Averil