Last year I planted a raspberry bush (?) in a pot on my balcony. This year some small ones started to sprout next to it. There is not enough space for more than one plant. Or will the parent die soon? They are also too close to the pot wall. What should I do now? Remove them or leave them? Destroy them? Relocate them (is that possible without hurting the parent plant?)?
The traditional way summer-fruiting raspberries were grown outside in the UK, the suckers are essential.
At the end of each season, you cut off all the old canes that have fruited, and the new canes (your suckers) flower and fruit the following year.
To grow them that way, you don't need to dig them up and replant them. The plant just renews itself with completely new growth every year, and any damage or disease in the old canes gets removed when they are pruned.
Summer fruiting varieties are the most common in the UK. However autumn fruiting raspberries produce fruit on the new canes, and you should cut down all the old canes in winter when the plants are dormant, ready for the new canes to appear in spring.
I recommend digging out the suckers and giving them away to friends, plus keeping some for yourself. Depending on the type, raspberries have a variety of ways they bear fruit. I am not sure you will continue to get fruit from the original plant in the pot because of the way that raspberries grow. This is a great article about raspberries and includes propagation information Fine Gardening.