We train to attempt a relight if there are no signs of damage, such as fire, seizure or vibration. It may be a simple flame out, though I'm not sure I'd still want to continue to destination.
For engine out acceleration we would level off and clean up once the engine was secured. We have a standard procedure and some airports have a special E/O routing due to terrain. With the special procedure level off is only done once established on the escape route.