Can/should I add water to the secondary?

I brewed a couple of batches yesterday with a few friends. It turns out that there was too much boil off and even tho I do full volume boils i ended up short. The yeast got pitched before I realized what was going on, so a 5 gallon batch with a target SG of 1.050 ended up being a 4+ gallon batch with a SG of 1.063. Can I add water when I rack it to the secondary after the primary fermentation is done to bring the batch volume up to where it should be?

Boil, cool, add.

I would be inclined to add it now.

You could add it now, or later if you do a secondary. As long as you boil (and cool) the water first, there should be no worries as long as you add the water carefully and minimize splashing.

Personally, if I feel the need to top up volume, to do it at secondary either with water or whatever suitable beer I have in cornies at the time.
Usually though, I rarely bother topping up unless its a beer that I know will be spending a long time in secondary (like my Burton, Scotch, or Strong Porter).

You can but I have never added it to the secondary. I think of that what Coors and Bud do to make their light crap!

Not a good idea to add volume to the secondary. If fermentation in the primary has started, don’t touch. If fermentation has not started you can safely add the extra volume of water IF it is boiled to to remove the oxygen and for sterilization, then cooled to the wort temperature. Don’t splash during the addition if you do go ahead with it. I would really suggest having a home brew and not worrying about the volume variation. Your OG being, sort of, higher than the target OG was most likely from the wort not being thoroughly mixed rather than an actual of 0.013 higher.

If I end up too high, I’ll chill and then add water before primary ferment. I avoid a secondary ferment as it’s not necessary in most cases, and can have adversely effect the quality of the finished beer.

I wouldn’t add water at this point as it may pose a threat to the quality of the beer. You’ll have better results letting the beer finish fermenting in it’s current state. The ABV will be higher than desired but the beer will benefit by it.

Thanks for all the great advice, everyone. Based on this feedback I will let it go as is. As one of my brewing buddies said, “Who knows? This might be the best beer you ever made.”