In a comment, John Lawler wrote:

It's an idiom, one of a number of them with the same structure NP + V + Adj, like He saw red, She went quiet, It runs slow. The meaning of this one is that the lamp will flash when you restart it, and the color of the flash in that case will be white. Oh, and if it matters, which it doesn't, white is an adjective here, not an adverb.

Stay strong!

Some commenters have suggested that white is a noun here, the direct object of the word flash, like the way you see happen with that verb when you flash someone (ahem!), or when you flash someone something. That’s nothing like what’s happening here. Flash is not acting as a transitive verb here.

Certainly color words can sometimes be nouns, including for special meanings like when you’re trying to separate your whites out for your laundry, or when your black print cartridge is running low so your prints don’t have good blacks anymore.

But here it is certainly not. You can prove this by inflecting it into the comparative and superlative degrees.

One light can flash white, but the next light can flash even whiter, and the third light can flash whitest of all.

Because nouns don't have comparative and superlative degrees, this cannot be a noun. QED.

This is a predicate adjective describing the subject, just like it is with is or seems or looks or becomes. Some verbs simply happen to accept adjectival complements; that doesn’t somehow turn those adjectives into adverbs.

Although as John mentioned, it doesn’t matter, to prove it an adjective not an adverb, consider that when the light flashes white, it’s the white light that’s doing the flashing. That shows that this instance of white is indeed an adjective.

Heading off more adverbial questions, I’m not sure what flashing whitely would mean; that seems borderline ungrammatical to me because it sounds more than a little bizarre to say that it is flashing in a white manner. I don’t believe color-words work as adverbs of manner, nor as adverbs at all. Flames that burn green aren’t “burning greenly”. I for one can’t say that and still have any idea what I’m talking about. It’s still an adjective.