Q: Can I add an author name in a published research paper?
As you can understand, this may be viewed a bit suspiciously, as (co)authorship is one of the commonest areas of dispute in research publications. However, if there is a genuine reason, you may certainly go ahead and give it a try. If the journal is a purely online one, it should be easier to make the change. However, if it’s a purely print one or a mix of print and online, it may be more challenging and take more time. You should write to the editor providing the reason and see what they advise and how to take this forward.
As there might be a fair amount of communication involved in this effort, you may find it worthwhile to invest in this handbook: A practical handbook of templates for communicating with the journal
You may also go through this related query about making changes in published papers: Is it okay to correct an error in a published paper?
Hope that helps. All the best for being able to make this change.