You can trim off the brown bits if you like, but I don't think I'd bother. The new growth will soon hide it.
It's quite small and looks as if it might have been pruned fairly hard and fairly late last year, possibly by previous owner trying to make the garden look tidy for sale. If so, that might explain both lack of flowering (it flowers on the previous year's growth and the time to prune is immediately after flowering, which is around midsummer here, mine's in full flower now) and the frost/cold damage (soft new growth following pruning is more vulnerable).
Doncaster, South Yorkshire. Soil type: sandy, well-drained
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