BSA ?
Correct! You've earned the point on this one.
The Hodge-Podge Special was created by Raymond and Elvin (sometimes listed as "Elwin") Poe, and campaigned by them beginning as early as April 1954 (Torrey Pines). The puzzle car was advertised for sale by E. Poe in April 1955, and subsequently raced by Dave Thormin under the name "Thormin Special" from early as November 1955 (Glendale) until at least November 1956 (Pomona). The Poes continued to race what you've shown to be a near-identical car under the name Hodge-Podge Special, also at least until November 1956. In fact, the two cars competed against each other (Thormin in his Thormin Special and Raymond Poe in the Hodge-Podge Special) at Pomona on 21 October 1956.
I'm marking this one as "Solved." While there's definitely more to be uncovered when it comes to these two cars*, I'm not going to keep this puzzle open for that.
Here is the ad as it appeared in the April 1955 issue of
West Coast Sports Car Journal, along with a color photo of Dave Thormin in his Thormin Special at the Santa Barbara Road Races (Santa Barbara Municipal Airport ) in early 1956, and the results from Pomona Road Races on 21 October 1956.
*
...such as, is it just coincidence that the car wears #14 in both photos, or are the photos from around the same time and I've somehow mixed up the dates?