Good idea - contact Friends of Clapham Common who are very proactive in trying to clean up the Common. They will appreciate more people contacting the council and will advise who to contact at Lambeth Council.
The woods have been cleared of some of the undergrowth recently, so it would be good to galvanise some action on cleaning them up.
The Friends organise occasional litter picks in conjunction with other set ups - Trash Heroes, Keep Britain Tidy and others. There was a person on Nextdoor who organised some litter picks on Sundays, but haven't see that for a while.
Currently there are over 60 members of FoCC who have volunteered to take on a small area each and litter pick - gloves and litter pickers provided by FoCC: https://claphamcommon.net/community-litter-pick/
Great exercise! The Common is definitely cleaner since this movement began early in lockdown, but it is an uphill battle.
It's worth people contacting Lambeth to complain as well:https://www.lambeth.gov.uk/forms/parks-enquiries