Whether to use flash on a close up shot is always a debatable issue, Carol. As a general rule, my default position is to start with a bit of flash then do a quick change if I am getting problems. But, I'm using more 'advanced' equipment and with an external flash unit. Some flash output compensation is normally essential. Can you vary the flash output level with your camera?
I will be shortly using my new camera, a Canon EOS Rebel SL1/100D, all I need now is the courage! Thank you so much Geoff for viewing, your time and advise, all duly noted. I cannot vary the flash output on my G7 which is now in Camera heaven, just thought I would upload a few from that camera before I use the new. I will try selective brightness reduction on this image but I doubt I'll manage that as new also to PP. As I recall this image was shot in AP mode. I am happy the eye is sharply focused.
If a subject appears to be likely to cause problems I like to take shots with and without flash just in case; but that is only possible when the subject is suitable. If I think I'm only going to get one shot then I tend to choose something which will be more on the under exposed side than over exposed, where all hope of recovery is lost. It is easier to brighten a slightly dark scene.
Having said that, I suspect you could still do a little bit of selective brightness reduction work on this image to balance out the brightness levels a little better.
The eye is sharply focused which is the main thing. Could you have used a narrower aperture to get more focus depth? I always shoot this sort of scene in aperture priority mode to give more focus depth. Slightly increasing the Iso can be helpful as long as it isn't too excessive so a very 'noisy' image will result.