Sounds like more of an ethical question than a management one. Who do you want to be? The honest guy who owns his (or his company's) mistakes and makes it right, or the shister who makes other people pay for his (or his company's) mistakes?

5% isn't a small mistake, but it may not be the end of the world. Is it larger or smaller than the selection margin for the contract? In other words, was the next lowest bidder between 400.000 and 420.000? If the answer is yes, then I think you should absorb it and make sure the estimating department never forgets about it. If you still would have been less than the next guy, then you could tell the client you made a mistake and try to negotiate payment. They may be willing to meet you part way.

Could also be a regional thing. You're in Italy? I don't have a clue about how contractors work in Italy or what would be considered normal practice in this case where you are. So keep that in mind.

A 5% overage should be a drop in the bucket between management reserves and profit, so long as the project stays reasonably on-budget otherwise this shouldn't be any big deal. Presenting this to the client would likely look worse bc you're concerned by the amount than bc you're asking for more money, so I wouldn't do it.

If the scope requested the work, and you forgot to include it in your price - you need to cover the overage.

If the scope didn't request the work, then the owner needs to cover the overage.

It is up to you to decide which situation you are facing and then be prepared to defend it to the owner if questioned.

the contract conditions is not quite clear.
under fidic red book,if the clarification is correct, and then the contractor doesn't quote accordingly, then the cost shall be deemed included in the rate. since the clarification is wrong, then the fault shall be cocurrent, such mistake shall negotiable .suggest you try to negotiate with the client for proper settlement of such cost.