Weeds growing through my bark chip path

Last year I covered all the grass in my garden with cardboard, and then put a thick layer of bark chips all over it. I then planted my new plants where I wanted them, leaving paths of bark chips all around them.

It looked beautiful (I am an amateur, so easily pleased).

However, this spring, weeds have started to grow all over the bark chips. So now, instead of grass I have weeds everywhere. Seems like a waste of lots and lots of bark chips.

I am keen to kill the weeds, and if at all possible kill them so they stay dead. I am often away from home for extended periods of time, so I am not really able to just pull out any weeds when they arrive. Naturally I would also like to keep the deliberately potted plants around the weeds to stay alive.

Does anybody have any tips on how to best achieve this please?