Hi,
Without looking at your National Insurance records and the S041 that has been issued to the Hungarian Authorities, I am unable to provide a specific answer to your query on this forum. Based on the information you have provided, I assume that there are periods shown on the S041 without any insurance or residency periods recorded.
If that is the case, and you are able to provide proof of residency, you can ask the Hungarian authorities to request another SO41 for those missing periods.
You can also request form CA9110 from HMRC which will provide you with a record of your Insurance and residence periods in the UK, you can show this to the Hungarian Authorities to support your case.
You can apply for a CA9110 by completing form CA3916 which can be found here -
Get a PDU1 certificate or a National Insurance contributions statement
Please make sure you attach a covering letter stating that you are requesting a CA9110 and not a U1 or Statement.
If you have any further queries relating to this, you can contact the National Insurance helpline on 0300 200 2500. Lines are open from 8am to 6pm, Monday to Friday.
Thank you.