Aqua Mail's notifications already have action buttons - but they only show if a notification is "about" fewer than 5 messages. It wouldn't make sense to allow the user to delete messages that he/she can't preview in the notification ("delete 25 messages who knows what they are").
Two things you can do:
- Mark your old messages as read, using Aqua Mail (open a folder -> menu -> mark read) or web mail
- Or set Aqua to notify you about "new unread" not "all unread" messages, this is in app settings -> message notifications.
In addition, if the phone runs Android 7 or newer, you can enable app settings -> message notifications -> smart watch enhancements, and then ...
... you'll be able to "pull apart" (vertically) Aqua Mail's notifications and see individual "cards", one per message, each with its own action buttons (but you will lose actions buttons on the overall notification, an Android limitation).
Also worth mentioning - lock screen notifications are not a "separate thing", they're just same "notifications" that Android displays in two places, in the status bar and on the lock screen (and a third place, an Android Wear smart watch if you use one).