A spavined horse is a lame one, especially a permanently lame one. The term has an interesting origin.
It's actually French. When a horse has a bad hock injury or bad arthritis in the hock, their gait behind starts to resemble that of a hopping sparrow. The French word for sparrow is "espavain."
Hence, a "spavined" horse is one that moves like a sparrow - and as it's most often associated with arthritis in the hock, it's sometimes used as a term for any horse that has a serious or long term injury that affects its ability to work.