can you advise what to do now?
You have three options.
1). You can manually forward those downloaded emails that are only in the mobile app one at a time back to your own email address which then eM Client for desktop will be able to download.
Note: If you do that option, make sure your eM Client desktop POP account setup is also set as I mentioned to "leave a copy of messages on server for X days u set eg: 7 days.
Or
2). You can select all those emails downloaded in the mobile app “by holding your finger on the first email and then just touching the other messages till all selected”. Then “once all the messages are selected”, press the 3 vertical dots at the top right and select “Save to file” on the dropdown. Those saved emails will go into your “Android or Apple Downloads” folder and will be all separate .eml message files.
You can then “connect your mobile phone to your computer” via a USB cable or WiFi connection and then drag all those saved downloaded .eml message files to a folder on your computer desktop. Lastly drag the .eml message file’s directly into your eM Client Inbox.
Note: If you have an iPhone, the downloaded files can be in your iCloud downloads which you can then access from your computer directly.
3). Lastly you could alternatively ask your mailbox provider if they keep backups of your mail which they then possibly might be able to restore. Some providors do keep backups.
As @ShaunY advised for future, you should really look at ultimately changing to an eg: IMAP, or Exchange etc type setup due to its advantages where all devices are allways synced both ways.