NASA and the USSR did many powered rocket landings, actually. NASA even had a program for reusable first stages via powered landings that was scrapped because automatic guidance wasn't feasible yet.
Failing to restart an engine in a retrograde trajectory. A difficult task that both the Soviets and NASA (and Arianespace and the Chinese) had issues with, but worked out.
It's very difficult because of the turbulent airflow in an engine bell pointed against the airspeed vector. Despite this, many rocket engines can do it reliably.
There are a lot of reasons for which an engine would fail to start in such a scenario. They all boil down to the turbulent airflow and high pressure in the nozzle inhibiting the ignition process.
How exactly you decide to fix it depends on many things. It's impossible to tell. But they are all downstream of the same failure, which is overcoming the high pressure and turbulent flow.
