Hi - My best guess is this is sap (phloem). When the plant had leaves on it last year, it photosynthesized and produce carboyhydrates and sugars that were used for plant growth but also extra is sent down to the plant roots so that in spring, when the plant begins new growth, it has the food/energy it needs to produce new leaves. So sap in some plants runs in the spring when pruned. Often it is in response to warmer days and cooler nights, which we have been experiencing. This shouldn't hurt the plant but I noticed you left stubs. Usually you want to prune to a side shoot or bud - about half inch above it. I can't really tell by the picture but it looks like your cut is several inches above the bud. 

Actually I was referring to that little bud on the left in the other picture. I don't know if there are any other buds above it on the backside of the picture. If you are taking down a whole stem, you would prune to 3 to 6 inches from the ground. When you are heading back, which it looks like you are doing, you would prune to a lower branch or bud and not leave a stub. I don't see any buds on the picture with the red markings. It could be that I just can't see them. Are there buds on the small stub? If not, then it should be pruned back at an angle to about half inch from the larger stem. As far as the larger stem goes, are there buds below that cut? If not, cut it down to 3 to 6 inches from the ground. 

Hi! 

Thank you! So this hydrangea is a good size and about 3-4 ft off the ground. I've attachd a better picture!

Ideally all your cuts would be about a half inch on a slant above a bud growing in the direction you want it to. You might need to see where the new sprouts are developing and then if there is a long stub above them, cut it back. Also, pruning often causes adventitious buds (that you can't really see) to sprout so you can see if any of these develop. Again I can't really tell from your picture where all the new buds are but the ultimate goal would be not to leave a bunch of stubs.