You could leave the cardboard there and just cover it with some sawdust or wood shavings (light to carry and rot down more quickly, if you can find a friendly wood worker) rather than chips, idea being eventually to have paths of bare earth as that all rots in, just kept weed free with hoing and a trowel.
Uisng plastic needs care as soil and compost tends to accumulate on top, then weeds germinate in that and can be tricky to remove. Again, wood shavings and sawdust are better than chips. I hope you back gets better too.