They can do whatever they want with it but you have no obligation to maintain nor support it if it’s not a documented public API, in my opinion.
It’s a bit like a house on a corner with a big front yard. People may cut through the grass to save time but you can’t blame the homeowner when he finally puts up a fence.
It’s unlikely that allowing the public to short-cut across your yard would create a prescriptive easement, but there are certain circumstances where allowing a party or the public to cross a parcel of land for a sufficiently long period of time does prevent the property owner from erecting a fence.
Although it’s not a hard-and-fast rule, in Ontario certain property owners have paths open to the public most of the time, but close them for at least one day of the year, often Christmas or New Year’s Day. The intent is to prevent access for any one or more continuous years, which in turn prevents an easement from being asserted by any party.
Although it’s far from as simple as, “if you allow the public access to your land for one continuous year, you allow it forver,” a certain folklore around this has arisen, and thus the practice.
It’s a bit like a house on a corner with a big front yard. People may cut through the grass to save time but you can’t blame the homeowner when he finally puts up a fence.