Thank you to all who have replied here. I am reassured that my anxieties are not seen as unreasonable even by the dog owners here. It's good to know that turning away is probably the most sensible option. I have pointed out to people that the notice on the gate clearly says dogs should be kept on leads, but as some here have observed, not all dog owners are responsible, sensible or courteous.
The gates were closed for an extended period last year after one owner refused to put her two dogs on leads while the groundskeepers were actually cutting the grass and there was a bit of a barney.
Children used to play in there a lot, and it was a joy to hear them, but I think the frequent presence of the dogs and occasional faeces has put them off, as I don't see so many now.
After your encouragement here, I will be more assertive with the offending dog owners and hopefully they will get the message and the space can be enjoyed by everyone again.
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A quote for those blessed with a spouse:


) and wait for the owner to control their dog (if I'm lucky). But in that time, I often have to endure the slobbery and sometimes snappy attention of their dogs.