Thank you for this response.
It's good that your children know their boundaries as it is important to teach them to really only allow the cat to approach them on the cat's terms, rather than them approaching the cat. Most cats do not like cuddles or being held at all.
The fact that the attacks seem to happen in one place may mean that he sees that as his territory. It would be important to try and exclude him from this area for now. Perhaps this can be achieved by blocking the area with furniture or using a stair gate/fencing. By keeping him away from this area, we may be able to stop the behaviour a lot more easily.
The fact that he breaks skin means it's likely this is more than play.
Be sure you are reading his body language at all times. At the first sense of a bite coming on, remove yourself and/or your kids from the situation. Watch for ears down, freezing, tail wagging etc. Some cats give lots of clues, while others do not. Then , give him plenty of space.
I am a big advocate of 'time out', whereby after a bite no-one acknowledges him for 5-10 minutes (no-one looks in his direction, talks to him or touches him). This lack of social contact should be enough to inform him that his behaviour was unwelcome. If we go down the route of punishing, such as pushing, shouting or scruffing, sometimes this can worsen the behaviouor as the cat can see this as rough play and enjoy the extra attention. For some, they have attacked because they want attention so we must not provide attention for undesired behaviour.
It is useful to build his confidence and sense of security in the home so he feels less inclined to lash out. This is done by enriching his environment with interactive toys such as laser pointers and battery operated mice, as well as food puzzles, cat mazes and cat tress. Some cats also enjoy catnip toys. Be cautious to not use your hand to play at any time, as this can increasing hand biting tendencies. Not providing enough structured play is the number one reason for 'attacking' behaviour and at his age he will want to play for several hours a day.
Another consideration would be stress-reducing products such as calming supplements in his food each day (e.g. nutracalm or Yucalm) as well as a Feliway plug-in. These can take a few weeks to kick in but work very well for some.
If these measures do not help, I would advise having a vet behaviourist come to the home; they can provide tailored plans and are often worth their weight in gold.
A final consideration would be to have a vet check if he has not had one recently. This would just be a nose to tail check to ensure we are not missing a source of discomfort or discontent such as a bad tooth or any joint pain.