The clay back to our fireplace is cracked in several places and one or two small pieces have flaked off.

It's around 100 years old so ripping it out and fitting a new one isn't really an option.

Is it likely that by using plenty of fireclay I can seal the cracks and fill the flaked sections or is it possible to fit a metal fireback over the clay one?

There are various fire cements avaialble and it's not massively expensive. If replacing is not an option then it might be worth a go.
You might find that thin areas won't stick too well though.

I'll give the fire cement a try first, then.

The fire surround and hearth would need to come out before we could replace the fireback and I'm concerned they wouldn't survive the experience, having been in situ as long as the fire.

all pics should be rotated 90 degrees clockwise, sorry about that.

Well I gave it some more thought and had a chat with a friend who's also a builder. His opinion was that I could patch it up with fire clay but it would probably only last a year at best.

Instead I've decided to have it relined with new fire bricks. He says he can remove the old fireback and build the new one without having to remove the hearth or fire surround/mantlepiece.

Hopefully have it done within a week.

the area I had to replace was 9x9 inches.
I bought two fire bricks measuring 9x4.5 and will cement them in tonight using 5:1 mix of fireclay to portland cement.
BTW my supplier has stocks of full firebacks, so probably not as rare as hens' teeth as someone on another post suggested.

Will be a good opportunity for you to check/replace the firecord between the edge of the fireback and the surround

Well, the builder did a very nice job of it.

Photo below.