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Adjust the Regular Expressions to deal with the 'missing' quotes in the html output:Replace:
If that doesn't work, can you give me (super) admin so I can take a look?
And tell me on what url I can see/reproduce the issue.
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The links seemed to have weird whitespace in them (before and after the href) and also would have quotes on some pageloads.

Anyway, this works:

Yes, you can do:And replace with:But use the "Between" strings to limit the replacement to only your menu.

You can use the Conditions to make it only be done on the pages you want.
Either via URL or via the menu items or any of the other Rule types.

not by css class. But you can use the 'Between' strings to limit it to the area you need.
If you cannot find beginning and ending html that is unique enough, you could use a template override to - for instance - wrap your menu in something unique like:

You should have enough information to do what you want/can with ReReplacer Pro.

You can search and replace using dynamic Regular Expressions.
If you need to only replace certain links - in your case, you will need something unique in the html you can limit the search by.
You can limit the Regular Expressions search itself.
And/or use the Between fields to limit it to a certain section on the page.
And/or use conditions to limit the replacement to certain pages/situations.

If there is nothing really unique about specific elements you want to target, you will have to use template overrides to add some unique html around or inside the elements.

Maybe possible, but not easy. You would need to use PHP to read out the current url and compare it to whatever you want to replace.