Do you have any research that backs up the benefit of having white noise on all night as opposed to just at bedtime?
There are quite a few studies showing benefits of white noise (mostly with very small samples of children), but a lot only seem to use white noise at bedtime, and I couldn’t find any studies comparing bedtime versus all night use – so empirically we don’t know with any certainty. White noise has been found to reduce night wakings when left on all night.

Theoretically – White noise is proposed to work in 3 main ways:
1. Reduces arousal thus helping children to sleep.
2. Blocks out noises that could be disturbing thus helping children to sleep, and preventing disturbances.
3. Becomes associated with falling asleep.

All three explain why it could be helpful at bedtime. Point 1 suggests that white noise overnight might be useful if it helps prevent your child from fully waking (which of course it may not depending on the source of arousal). Point 2 suggests that white noise overnight might help if your child is prone to waking in response to sounds in your environment – if your child seems to sleep through environmental noise – white noise overnight may not be necessary. Using white noise overnight would actually work against Point 3 because white noise would presumable precede falling asleep and waking up, therefore diluting the association with falling asleep.

My feeling based on the above is that in most situations white noise is probably most useful at bedtime, and possibly until you go to bed (to mask any noises you make), but only useful beyond that if your child seems to wake frequently at night in response to noise disturbances.

Research is helpful as a guide – but ultimately, the best test is whether it works for your child, because all children are different.

Drawn from the following review and linked articles:
France, K. G., McLay, L. K., Hunter, J. E., & France, M. L. (2018). Empirical research evaluating the effects of non-traditional approaches to enhancing sleep in typical and clinical children and young people. Sleep medicine reviews, 39, 69-81.