As you can see from the username I have a cockerpoo. I know lots of people slate them,I got mine for a range of reasons and have actually lucked out but we have worked really hard on training and I did a lot of research and made sure both her sire and dam were fully health tested and from good lines.
She has her challenges, she is too clever for her own good and if you don’t satisfy her needs she can be difficult, but she is very loving and our life works well & I wouldn’t be without her.
I have known cairns & that cross has the potential to be very difficult, as PPs have said cairns are very stubborn dogs ( they are terriers) and with a poodle intelligence you have the potential to end sup with a very challenging dog.
i would second a mini poodle (not toy), mini schnauzer are lovely dogs ( but very barky). But worth thinking about whether you’re really looking for a toy/house dog or something more rough and tumble. I think based on your life it’s probably the former. Also think about grooming needs, it costs about £50 every 6/8 weeks for mine to be groomed and she really needs brushing every day, it’s not for the faint hearted. Also think about whether you will need to leave the dog alone and for how long as many toy/companion breeds suffer from separation anxiety (& any dog needs building up to being left alone).
Finally bear in mind many small dogs are terriers, which are by nations stubborn, tenacious, confident and independent- they had to be to survive the jobs they were bred for, so are not going to be chill, quiet, easygoing etc, so you need to think carefully about what will fit with your family and the personality of your kids.
The kennel club breed finder allows you to refine by characteristic so might be a good place to start.
Good luck though, it’s incredibly rewarding.