It is easier to think of "Jira Core" as being "a platform that provides a lot of issue tracking functions, and the ability to add things which can give you a lot more functions on top of that"
Jira Service Desk is a "Jira Application" that adds a lot more functions. The first step for you is to log in as an admin and go to Admin -> Manage Applications. This tries to help you look after the applications you might have added, and should show you the current version, and a button or link that leads to upgrading the current one. Assuming that "core" supports the newer version.
If that shows you can upgrade the application, then the best (and supported) way to upgrade is to click the upgrade application option.
If it does not show you an upgrade option, then either there is no upgrade available, or, more likely, there is, but your version of Jira Core cannot support it. In this case, yes, you need to upgrade Core first.
Introducing Jira Cloud for Excel Here at the product integrations team at Atlassian, we are thrilled to announce the new Jira Cloud for Excel add-in! This add-in lets you export Jira data directly ...
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