the oozing goo looks like phytophthora to me, but I have never seen it on a fresh green branch like that. I'll wait for somebody else to comment.
If it is phytophthora you can treat with Monterey "agri-fos/garden-fos" phosphorous acid. I have had it before and this treatment worked well for me.
Pretty darn sure Brian is correct. I have had this on my kumquat and Tahoe Gold mandarin (on old and new shoots). In my case, it was all my fault!
During a period of switching out the flagging tape for the tanglefoot on both trees, ants starting farming black scale. I got busy for a couple weeks and came back to find both trees infested with the damn scale. So I squished all the ones I could see with my finger nail. I figured I'd spray 'em down right before reapplying the tanglefoot. Within a week, I noticed the same hard oozing coming out of the general area where I killed the scale. I'm pretty sure I also created a place for the spores to enter.
It did actually damage both small trees pretty bad. The green fresh shoots ended up dying and the older, more woody pieces died back considerably. I lost all the mandarins, too. Luckily they are both growing like weeds now. I did not treat it with anything. I just severely pruned my trees (little bushes). Lesson learned.
I think you may have had something nick your tree, which provided a place for it to enter, but it was not severe. I personally would cut it off so it cannot spread.