The sprouts aren't a problem, other than that they are fragile. If you are planting by hand, no big deal. You can also just trim them back, but be aware that if you use the same cutting tool on different potatoes without sterilizing, that is an excellent way to spread disease.
When a potato has many long sprouts, that is an indication of increased physiological age. The first sign that tuber aging is becoming a problem is when they start to develop sprouts all over the tuber, rather than just at the distal/rose end. This indicates a loss of apical dominance, which is the first sign of hormonal changes due to tuber age. Plants grown from tubers with increased physiological age tend not to yield as well and also produce more small tubers due to the increased number of stems. The difference may not be so much that you notice, but it is generally something that you want to avoid. If you often have long sprouts on your potatoes by planting time, it would be a good idea to find a way to store them colder.
The mother tuber only lasts a year, so there is no point in replanting it. They do sometimes remain in very good condition, but they won't grow again.