Hi Rachel,
I'm glad you included an image of the splitting bark as that is likely the issue. Splitting bark weeping sap on a white pine indicates a canker of some sort. Cankers can be fungal or bacterial in nature, and the pathogen infects the living tissue of the tree limiting the ability of the tree to move water and nutrients through the tree (causing half yellow needles in this case). At this point, the tree is suffering from drought like conditions due to the pathogen. 
There is no cure for a canker. Because the tree is small (and not a threat to structure or life) you may wait and see what happens this summer. It it likely the tree will die; waiting may make you feel more comfortable with the situation. 

One note is to watch for yellow blisters near the canker that release orange spores. That indicates white pine blister rust, which can occur when white pine is growing near gooseberry, or ribes species. If white pine blister rust caused this canker on the white pine, you may want to choose another species when replanting. You can read all about white pine blister rust at the link below. Might be good to compare symptoms listed in the article with what you are seeing on the tree. 
https://extension.umn.edu/plant-diseases/white-pine-blister-rust

I hope this helps. Please let me know if you have any questions. 
Thank you,
-Julie