I took a photo against the sun and got some nasty orange artifacts:
Going black & white makes it less annoying, but I don't like the overall image. Are there any other post-processing techniques to mitigate it?
A lens hood most definitely would have solved your problem. But, in the interest of helping you with this photo...
Your question is tagged Lightroom but it really needs to be tagged with Photoshop as well. I don't know of any solution in LR to solving the flare issue. I've done some cursory edits to your image, details below:
The orange spot flair on Gent's neck was removed with the patch tool - which you may be able to simulate in LR.
The flair on Gent's face was removed using a combo of:
All of the above layers were Gaus. Blurred and opacity reduced to some degree to aid in the blending of the adjustment.
I stopped working on this before getting to Little One's hair and the back of the neck because, I think, you get the point.
The flair can be completely removed, or downplayed more...depends on how much you want to edit. Either way - I think you need to dive into Photoshop to meet your request.
The issue you have with that image looks like it would have been solved with a simple lens hood. Depending on the lens you use you can find branded options and off-brand options (which are generally cheaper but normally of the same quality). They also have the added benefit of keeping your lend more protected in the event it's dropped. Some additional information related to lens hoods in regard to your issue:
What good is the hood? The primary use of a lens hood is to prevent light from hitting the front lens element from the sides - reducing contrast and creating flare. Pictures taken with a lens hood installed can have richer colors and deeper saturation.