I have a number of emails under the heading Gmail for which I cannot find a way of deleting, any ideas please?

Please describe in detail all you have attempted to do in order to delete your mail.

In mail, unless your preferences have specified otherwise, messages that you have tried to "delete" will still be there. In order to remove the messages completely, you first select the messages you don't want, right click, click "delete" and they will disappear from your inbox, but only for a while. After this, you right click on your mailbox in the sidebar and click "erase deleted items". This will completely remove the messages your deleted.

Happy new year :)