How do I get paid for extra works that come up on site?

Short answer

Hold the extra work until there is a written instruction ordering it.
Ask the designer for a site instruction if the extra changes the technical solution.
Price the work by cost-estimate line, with quantities and unit rates, not as a lump sum.
Sign the contract addendum or additional order note before you build it.
Issue the work statement for extras separately, so it doesn't blend into the original budget.

Civil Code, Art. 1270(2) — a contract is amended by agreement of the parties; extra work built without an addendum is a claim that is hard to prove.
Public Procurement Law 98/2016, Art. 221(1) — additional works that have become strictly necessary are documented by addendum, and the price increase may not exceed 50% of the original contract value.
Law 10/1995 on construction quality — departures from the verified technical design require the designer's written agreement, issued as a site instruction.
Fiscal Code, Art. 281(7) — for construction and assembly works the taxable event arises on issuing the work statement or, per the contract, on the client accepting it.

Example: excavation uncovers an old foundation the geotechnical survey missed. Demolition and backfill are priced at 34 m³ of concrete broken out and 41 m³ of compacted fill, i.e. 18,600 RON. The additional order note is signed, then the work proceeds; skip that step and the 18,600 RON stays with the contractor.
Example: the client asks for a partition wall to be moved after services have been run. This is not remedial work, so it is quoted separately — 3,200 RON labour and materials plus 900 RON to redo the services — and accepted in writing before anything is taken down.

Building on a verbal request made on site, then negotiating the price once the work is done.
Presenting the extra as a lump sum with no quantities — the client disputes exactly the part you cannot evidence.
Mixing extras into the original budget, so you can no longer show the overrun didn't come from execution.
Skipping the designer's site instruction on a change of technical solution, which then blocks handover.

Extra works are recorded as distinct stages with their own budget, separate from the project's original budget.
The documents ordering them stay attached to the stage, so at handover you're not hunting through email.
The work statement for extras is generated separately, from the stages confirmed in the journal.
The budget dashboard shows the original budget and the extras on separate lines.