I set up a googlemail account on a computer, using the existing gmail account on an i-phone. Although the googlemail account works on the computer.... the gmail account on the i-phone can only receive email, but no longer sends them ?

You probably did not enter the outgoing server settings correctly. There are separate settings for incoming and outgoing, and your gmail ID and password must be entered twice, once for incoming and once for outgoing.

You might want to consider using the gmail app instead of the iPhone's built in mail client. Google has started making it harder to use other email clients, warning you that they may not be secure, and requiring you to change a setting that says essentially that you will assume any risk.