Summing up digits manually takes too long. A decent strategy seems to be: start chaining a long path together. At some point it will pop by itself, or you will notice what's needed to complete it. Suggestions: There should be a button to clear the chain in progress because you can box yourself into a dead end. A pause button would also be nice (maybe blank the screen when paused).
Randomly chaining numbers seems to be the most practical strategy up until you start running into the empty blocks. Game should probably be made a little easier so the intended strategy of chaining primes becomes more viable.
I found that the easiest consistent strategy is to play it like Minesweeper: start with randomly clicking (to gain time) until you find it too risky, then start with the easy portions (2-3 chains like 5-5 and 3-6-1 to clear out blockers, also remove 1s and 2s to increase overall average value), then move on to the difficult part when you're ready (chaining 40+ numbers when you have a lot of time).
There are of course other strategies, but I'll leave that for others to discover. :D