Guessing from what you've said, I'd say dodgy hot-shoe contacts but let's go through the process of elimination.
D-Lite One is entry-level kit but well made and I'm not aware of any issues. I used them very happily for a few years before moving to Godox, primarily AD400Pros and mainly for battery-powered location working with more power and faster recycle.
D-Lite Ones have a slowish recycle time by today's standards, around 2 secs as I recall, but that's at full power and if you're at 1.0 that's minimum and should be very much faster and cope with rapid firing okay. That would also tend to rule out overheating issues though if you're using an enclosed softbox with the light pointed down, the tungsten modeling lamp can produce a lot of heat if you have it at full power.
So, switch on the beeper and try repeat firing using the test button on the head itself. If those comments about a softbox and modeling lamp running flat out ring a bell, replicate that situation. Press the button firmly as those little touch-pads can be a little uncertain, then give it some stick. If it fires 100% that would probably rule out the flash head as an issue.
Now do the same thing but firing the flash from the remote's test button. If it passes that check too, then it suggests something wrong at the camera end. It's unlikely to be the camera itself, but the hot-shoe contacts are a common weak spot. Shooting with the camera vertical in portrait mode puts more strain on the connection, and Sonys with their unusual hot-shoe design also comes up as an issue from time to time. Clean the contacts on both the camera and trigger carefully, make sure the trigger is properly seated and secured, then cross your fingers and see what happens as you replicate a hard shooting session.
Just had a look at your website Manda. Fine work, you have a great eye and your relaxing style mentioned by clients is clear to see. That's the real skill of portraiture, way more important than parabolic this and gridded that