I think people have been a bit harsh to the OP.
4 months old puppies can be very scary to people who are not used to dogs, the bounce around,nipe your clothes, jump up, bark and can be pretty full on.
A well trained adult dog is much easier to deal with.
OP I would discuss this with your Mum, what is she finding scary about the dog. If it is puppy behaviour then this will pass and as the dogs becomes calmer your Mum will gain more conficence (however this is not going to happen over night and will take some work from you)
On a personal level I would not consider a third child while I had a young puppy and would want that situation calmer before adding to the mix.
When your Mother visits keep control of the puppy maybe have on a lead to keep the dog calm.
I would also book a 121 with a trainer (who can advise over zoom). This will help to give you a programm that will work for your dog and your family. If your Mother could be involved and voice her concerns these can also be dealt with by the trainer.
It is hard to make sensible decisions when your are emotionally involved so do take professional advice - it may not be as judgey or catastrophic as here 