Put a large plastic bag as a liner.
A garbage bag or plastic grocery bag can serve as a liner, when full, just close, add the items to your larger bin for pick up and replace the liner in your wicker recycle basket. Easy peasey!
Scroll down to #4: https://www.bobvila.com/slideshow/wicker-gone-wild-10-easy-ways-to-transform-old-into-new-44469#diy-outdoor-furniture
Just add a garbage bag. You can find smaller kitchen garbage bags that fit perfectly or just use old shopping bags.