We have acronyms for main engine cutoff (MECO) and secondary engine cutoff (SECO). Do we have an acronym for secondary engine startup---SESU maybe?
Since you tagged this with falcon-9, we can simply look at the webcasts and the mission timelines SpaceX publishes. Second stage engine start is called SES if there is only one, and SES-1, SES-2, etc. if there is more than one.
The first one, however, is normally not called out. It is simply part of the stage separation sequence. So, you will never actually hear "SES" or "SES-1" called out explicitly. You will only hear "SECO" if there is only one firing of the second stage, or "SECO-1", "SES-2", and "SECO-2" if there is a relight.
So, technically speaking, the answer should really be "There is no such term for the first firing of the engine", however since the second firing of the engine is called SES-2, we can infer that the first is probably SES-1 or simply SES if there is no SES-2.
Here is an example from the live stream for the SiriusXM SXM-7 mission. I cued up the video right before SES-2 and SECO-2:
However, this only applies to SpaceX. Other launch providers use different terminology, for example BECO instead of MECO.