Sounds more like a job for Photoshop.
You could use the Lightroom clone tool to try to clone a piece of the wall that looks good onto the area of the wall with the flash, but I have had little success with this type of cloning (I suspect its a limitation of either the user ... me ... or a limitation of the tool)
Flash reflecting off the background is one of those gotchas that is difficult to fix. For me, it's even difficult to do it in Photoshop. The real key is to use bounce flash, or angle yourself so that you're not shooting directly against the background. Easy to say after-the-fact, I know. I tried fixing a photo of my daughter and granddaughters that had a glaring reflection from the wall. I tried several different approaches that ended up looking worse than the reflection. But then, I'm not a real "high end" photo finisher when it comes to those types of problems.