We have had quite a long and cold winter, with temperatures often below freezing point. We also had frost and what I would like to know from the attached picture, do you think a tree in this state will fully recover over time?
I don't see why not if the scion is still alive. How was it affected by the frost? Is there still foliage above the graft point? By the way, the growth seen in the photo appears to be from the rootstock. It should be removed if that is the case.
No foliage above the graft point. There are only a few trees left with foliage above the graft, most of the trees have only growth from the rootstock and will be removed.
Scratch the bark on the scion and see if it's green underneath. There's a chance the tree will make a comeback if it is.
Since you asked (via PM), yes, grafting new scions onto surviving rootstocks is an option if you have the expertise to do so. There are a number of different techniques that can be used.
Very good news indeed, is it plenty of expertise that is needed? And what techniques would you suggest? Some of the rootstocks are very healthy and have quite thickish rootstocks whilst others are younger and thinner.
I'm not an expert on the subject so I'll leave it to the following document you get you started: HS1309/HS1309: Citrus Propagation.
Grafting is not so difficult as it may seem. Just watch some videos from Youtube, and try. Experts have just higher success rate than novice, but you can't become an expert without starting as novice.
Irrigation When leaves are lost, transpiration from the tree canopy is greatly reduced. Therefore, the amount of water required should be reduced. Excessive irrigation does not result in rapid recovery, but may result in root damage and nutrient movement below the root zone. Normal irrigation should be practiced when trees regain their normal foliage development and canopy density. Irrigation in the winter following cold injury should be reduced because it may induce new vegetative growth, which might be damaged by subsequent freezes. If freeze damage occurs early in the winter, cutting back on irrigation will delay tree growth until the danger of additional freezes has passed. However, trees that put forth new growth should not be allowed to get water-stressed.