I know nothing about UOF in mental clinics so bear that in mind with my answer.

I would have thought that use could classify factors that affect UOF as those that remain with a staff member between 2014 and 2015 (e.g. how violent they are) and things which don't (e.g. essentially random factors, a staff member might treat their same patient the same way and sometimes they require UOF and sometimes they don't). Some factors like the section they work in might be the same for some staff but not others.

In a single factor model you are assuming that there are factors due to the staff member plus a random element.

The staff members who have the most UOF instances in 2014 probably are more likely to have UOF in 2015 but also probably had higher than expected levels by chance.

To simplify things say staff members fall into two types violent and non violent. violent types have a number of UOF which is (approximates) normally distributed with mean 10 and s.d. 4, non violent types have a mean of 4 and a s.d. of 2. The staff member with the most UOF might have had 30 he is almost certainly of the violent type but it was just random that he exceeded the average of violent types of 10. The expected number of UoF the following year for that staff member would still be 10 (or a little lower as there is a small chance he is non-violent). An R squared of 0.21 implies that the correlation is not very strong and the random (or unmodelled) factors have a greater effect on the dependent variable than the independent variable.

Of course all staff have a different mean level but it makes sense that the staff with least UoF were not only less violent but also lucky (if UOF incidents is considered a bad thing) and those with the most were not only more violnet but also unlucky. This assumption mean that for linear model of

UOF15 = a + b* UOF14 I would expect a to be positive (someone with 0 UOF in 2014 has an expected value greater than 0 in 2015 and b <1 (If the total number if UOF is similar in each year the people with the most in 2014 would be expected to have more than average in 2015 but less than they had in 2015).

All that said for the regression model to be reasonable the residuals need to be independent, normal random variables. If the relationship is non linear then the residuals (difference between the actual and predicted 2015 values) will not be independent. If the actual relationship is quadratic most of the staff who had very low UOF or very high UOF in 2014 will have negative residuals and those in the middle will have positive residuals (or vice versa). This is easiest just to plot and do by eye but if you have enough data you can put them into groups based on 2014 UOF and test whether the mean residual in each group is close enough to 0 to have occurred by chance.

Whether to include a factor or not an exact science if you have few data points you don't want many factors in the model, you also need to consider whether while the quadratic improves the relationship a little another function might improve it move.

With your data I would consider taking logs. Without taking logs your residuals have a floor (someone who had 1 UOF in 2014 can not have a residual less than -1) and I get the impression from your post that the standard deviation of the residuals will be high enough that the normal assumption of residuals would give many staff a significant likelihood of negative UOF in 2015.

Taking logs would usually make your model

Ln(UOF15) = Ln(UOF14) +c

but in your example you can have 0 UOF just not negative where a log based model assumes all the variables are positive. I don't see why you couldn't change the model to:

Ln(UOF15 + 1) = Ln (UOF14 + 1) + c if you do have 0 data.