Can you get access to Gales Digital Newspapers (English Collection 1600 - 1900)? I just did a very brief search there for "lashmar AND jockey", and got a few results, though his(?) death in Lyons in July 1890, as a result of a race fall, is reported as Thomas Lashmar.
Les
PS: Lashmar competed for the Prix du Grand Camp, Lyons. The horse was "Naaman", which kicked him in the head, and he did not recover.