sprinkler pressure issues
Reason

A sprinkler head's pop-up mechanism seals and becomes water tight when fully raised/extended. This only happens after being supplied with sufficient water.

A sprinkler that is not fully extended will allow more water to exit than the same sprinkler that is fully extended. The high water pressure actually creates its own water tight seal, and only allows water to escape through the spray head. Low pressure water will escape from around the pop-up mechanism and the spray head. 

The catch-22 is that with too low of pressure, the system can't self seal and build up pressure. It will do what you're seeing and just continue to never seal.

A way to test that I'm correct is to get a small group of people and have one person stand over each sprinkler head and pull it up while the system is showing these symptoms. Fully extending all of the sprinklers manually and holding them there will seal the system and make it start working on it's own. They could all now walk away and the system will continue to work until it is shut off. 

After my explanation, it may seem like I'm saying that your problem is low pressure. Your problem is actually low flow. Water pressure only exists because of a restriction to flow. If you cut off the end of your water main after your water meter and let the water freely shoot towards the sky, you'd have 10 gallons per minute of water going everywhere, but your water pressure would actually be at 0 psi. Water pressure doesn't mean much without knowing the amount of flow as well.

It's confusing because your sprinklers seal from both flow and pressure in a way. They seal when the restriction to flow is great enough to raise the heads and make them stay there. If you don't have enough flow, you'll never build up enough pressure to overcome the leaks around the pop-up mechanism for it to self seal. 

If we can increase the flow at startup to this zone, it will make the sprinklers seal. Or if we can decrease the number of sprinkler heads on this zone, we can reduce the flow required to make the the sprinklers seal. 

Solution

The solution requires you to find out why this one zone is different from the others. Does this zone have more sprinkler heads than the others? Reduce the number of sprinklers on that zone.

Is this zone at the end of a long run of a copper pipe that's too small of diameter, or a galvanized steel pipe that's corroded and restricts flow inside? Install a larger diameter pipe from your main to your valve.

Is your sprinkler solenoid valve only 3/4"? Increasing it to 1" might be all you need to do. The sprinkler solenoid valve is typically the largest bottleneck in the entire system. 

Splitting your misbehaving zone into two zones is an option, as that will cut the flow required to self seal in half. 

If you can't do anything significant like that, your final option is to replace your sprinklers with low flow sprinklers with the same coverage. You'll need to water for a longer duration, but it'll reduce the flow required to self seal the system at startup. 

Conclusion

Anything that will increase flow to this zone and removes bottlenecks, or that reduces the amount of sprinkler heads or the sprinkler head flow will be how you fix this problem.