Four years ago, as Andrew Cuomo prepared to begin his first term as governor, the biggest problem facing New York state was how to close a $10 billion budget gap.

This year, gearing up for his second term, Cuomo faces quite a different challenge: what to do with roughly $5 billion in extra cash.

This windfall — not really a “surplus,” in the usual sense of the word — comes from an extraordinary flurry of penalties on major financial institutions accused of breaking federal and state laws.

The biggest chunk, $3.6 billion, will come from the French bank BNP Paribas, for violating US sanctions against Sudan, Iran and Cuba.

So how should state leaders use the money?

The best general answer came from Assembly Speaker Sheldon Silver. “It’s a one-shot revenue, it should be a one-shot expenditure, and we should invest in infrastructure,” he said last month.

Cuomo and Senate Republican Leader Dean Skelos have taken similar positions.

The big question is how Albany’s big three will end up defining “infrastructure.”

For example, the latest five-year capital plan of the Metropolitan Transportation Authority has a $15 billion funding hole.

The state Department of Transportation will soon issue its own new five-year capital plan, likely with its own large funding hole for highways, bridges and upstate transit. The state Thruway Authority, stressed by construction of the Tappan Zee Bridge, is also tapped out on the capital side.

Combined, these needs are so great that, even if pared down to absolute essentials, the windfall would make only a small difference. But it can be a crucial one, if spent strategically.

Senate Republicans have focused on rebuilding local roads and bridges. They could accomplish that goal by devoting a chunk of the windfall to putting the state’s Consolidated Highway Improvement Program on a pay-as-you-go basis.

CHIP is now financed entirely by borrowing, with debt service approaching the program’s total annual outlays.

Although Silver hasn’t been more specific, mass transit is a traditional priority for Assembly Democrats when it comes to infrastructure.

Any windfall money targeted to the MTA capital plan could most effectively be targeted to “state of good repair” maintenance, too much of which is now financed with bond money.

The danger is that both houses of the Legislature will face intense pressure to spend the money on school-aid increases.

But school aid is a recurring expense, not a one-shot — which means lawmakers would have to find another source of funds (most likely, tax hikes) to avoid cutting school aid in later years.

Cuomo, to his credit, already has flatly rejected the idea.

In a financial-plan update in July, Cuomo’s budget office said the governor’s plan for spending the windfall “may include funding one-time capital expenses, bolstering reserves and reducing debt.”

More recently, however, Cuomo has suggested he’d like to divert up to $1.5 billion — nearly a third of the whole windfall — to an “upstate revitalization fund,” using his existing regional-economic-development program to “invest in catalytic infrastructure projects as well as quality of life initiatives.”

The fund’s investment would be modeled on his “Buffalo Billion” initiative, which has plowed taxpayer money into everything from stadium upgrades for the Buffalo Bills to a highly speculative $750 million investment in a solar-panel factory.

A better approach to upstate revitalization was modeled in a plan released last week by Syracuse Mayor Stephanie Miner.

If $1 billion was available to her city, Miner said, she’d devote most of it to rebuilding the aging municipal water system, augmented by an innovative chilled-water project that could reduce air-conditioning costs for businesses in Syracuse.

If the governor sets firm enough parameters, the Legislature will have little choice but to follow suit and turn back to basics — pay-as-you-go financing of basic infrastructure needs, not more corporate subsidies or political pork under the guise of “economic development.”

And certainly not more school-aid spending, which could only be sustained with future tax hikes.

While he’s cleared up the red ink that greeted him in 2011, there’s at least one way in which Cuomo’s fifth-year budget will be similar to his first-year spending plan. Both require the governor to say “no” to a lot of people — including, this time, himself.