Are those your pics, or stock photos of an eroding headcut?
Is your 12" siphon discharging at full bore? It should be releasing an incredible amount of water right now. If it is not, then you have some clogging in your siphon system. DO NOT GET CLOSE TO THE INLET NOW, THE SUCTION ON A 12" IS STRONGER THAN A HUMAN.
Yes, I agree you need to fix your emergency spillway right now. The force eroding the headcut is due to gravity moving the water. You need to get the water from the high elevation at the crest of your spillway down to the low elevation at the bottom of the headcut.
How much water flow do you estimate is going through the emergency spillway now? If it could pass through a 6" pipe, one option would be to buy some 6" flexible corrugated drain tile pipe and build a sandbag wall for the inlet side on narrowest stretch of flow along your emergency spillway. Pound some rebar into the ground in a v-shape as a debris guard. Run the outlet of the pipe down the headcut into a stilling pool.
If needed for a sufficient rate, you could probably add a second run of 6" pipe through your sandbag wall.
If your flow rate is much higher than that, then you probably need to move up to double-wall plastic culvert. Unsupported ells in culvert are difficult to maintain against high flows. You may have to install your culvert angled down from the top of the headcut down to the bottom. You will have to build a bigger headwall in that configuration (of sandbags or dirt) to direct your waterflows down the culvert.
If your current flows are bigger than that, then dumping a load of rock may be your best short-term fix. However, as Dave points out, that will slow the erosion rate of your headcut, not stop it entirely.
After you get past the emergency fix, you can come back with more pics and dimensions for a cost-effective repair when your ground is dry.
(Even if my suggestions don't work when you survey your specific situation, hopefully they help you create a workable solution.)
