Can I Move Cat And Kittens To A Safer Place?

It's a risky thing to do.

A lot depends on the mother cat, how well she knows and likes you. What her general temperament is like, and what her mood is like at the moment.

People do move cats and even very newborn kittens safely, but sometimes it goes badly and the mother may either A) abandon her kittens, or B) frantically try to move them back, harming them in the process... or even try to move them somewhere else.

The general advice is not to usually touch the kittens until 2 weeks old.

UNLESS you have a close a bond with the mother. In which case, you could try to show the mother a new nest. If that goes okay, try to move just one kitten there (handle the kitten for minimum time, and perhaps even use gloves when you do). See if the mother accepts this. If she doesn't get defensive or try to immediately move the kitten back, you could try to move any other kittens, too.

Still, if at all possible, best to leave the kittens where they are until at least 2 weeks old. Is there some way you can make their current nest safer?

Cats are extremely sensitive animals. They need a lot of attention and love from their owner. If you want to move your cat and kittens to a safer place, it depends on where you are keeping them now. If you think that their present shelter is a bit too cold, you can shift them inside the house. If they are staying outdoors or in your lawn, or if you live at a place where the climate is cold, you can keep your cat and kittens in your bedroom or living room. Help them settle in at night by placing warm blankets in their beds. A hot water bottle can help the cat and kittens stay warm. If you are concerned about saving the cat and kittens from other animals like dogs, keep them inside the house as cats don't prefer to live in cages. If you think that your cat's present home is too hot, make it sleep in a cooler place, preferably under the fan, or at some place which is airy, for example, a terrace. You should also keep plenty of water near them so that they are not dehydrated. Cats like roaming around, that's why their places should never be congested.
If you really want to move your cats and kittens to a safer place, you should keep the animal's habits in mind, and try to look after it as best as possible, because your pets are your responsibility.

It's all safe and sound if mother is not rejecting. You can move them easily by now and it will not effect either of them. But a problem can be there if they got familiar with this  place and then you will try and things will go in vain.