Domain names are handled by a set (not a list) of authoritative nameservers. A set means there is no inherent order.
This set is published in the parent zone so that the delegation works.
What it all means is that, under normal operation, each one of the authoritative nameserver will get the same share of DNS queries, at all time. This is primarily load balancing and not fault tolerance. What is true however is that recursive nameserver will learn the set of authoritative nameservers per domain, and will try to use the fastest one. If one does not reply at all, then they will switch to the other one in the sets. But this is still mostly per query, and hence will seriously hinder resolution.
The vocable "primary/secondary" is often used (or "master/slave") to denote that you basically control one of the nameserver in the set (the "primary") and then the other ones are just configured to feed them off the primary one. Which means, as long as you update only the primary, and waiting for "some time", all the secondaries become synchronized automatically.
This is only one way to operate, there are many others. Like for example having all of them being transferring content from an hidden master. Or all of them being at the same time configured from some external resource, like a database.
"I do not have control on the primary nameserver" is very strange, and really not a setup you want to be in. You first step should be to change that and change nameservers for something you control. Just use another provider, set up the zone with the exact same content as now (except for the NS records of course) and then go at the registrar for the domain and change the nameservers. After waiting some time, your new authoritative nameservers you control would be the only thing relevant and you will then be able to change the content of the zone with confidence and surety.
So, the act of changing something on the secondary should not happen, in the sense that they should be barriers to forbid you doing that.
Because contrary to what another answer was telling, there is absolutely no guarantee that things would resynchronize themselves automatically here. It all depends on how the setup is done and how you made the change (did you change the serial or not), and your question lacks a lot of these details.
It remains that, if all of your nameservers do not reply with the same content because they do not serve exactly the same zone anymore, it means that like, for 2 nameservers, 50% of the population will get the content as served from one, and the other 50% as served from the other.
This will create of course, at least a lot of confusion, and maybe breakage.
So the easy morale of the story: always ensure all your nameservers have the exact same zone to serve.