So what happens if public deficit (I interpret as government spending more than comes in?) is reduced? Your post seems to say it okay for gov to spend more than it takes in and leave less for the economy.
Let's say part of the government expending is maintaining the railroads. So the State has a number of people paid every month to go to the railroads, remove the old railway sleepers and tracks and replace them with new ones.
To do this the government also purchase the tracks and sleepers from private companies, who manufacture them and ship them to wherever they are needed.
One day, the government decides it needs to cut spending. And among the cuts, it decides to postpone the maintenance of railroads. The workers who replace the tracks get laid off and while they apply to unemployment subsidies, they can't consume as much as they did. So the restaurants which sold them food for lunch sell less food. Maybe they fire a waitress, or if one resigns, they don't replace her. Production falls at the companies making the tracks and sleepers. Maybe they postpone hiring new people. Maybe they suspend workers, depending on how hard they are hit. In any case, the workers there notice the fall in production and, if the overall economic situation in the country is bad, they fear they'll get laid off. So they cut back their spending and save their disposable income instead of consuming.
As this happens, the State income gets reduced. If there is VAT in the country, the fall in consumption means less VAT gets collected. Private income is getting reduced, either because people are getting laid off or because people aren't expending (so, if one of the workers at the track factory decides to postpone the purchase of a new computer, the salesman at the computer store looses a sale and makes a little less money), so less money enters the State through income tax as well.
Even more, railroad maintenance is getting skipped, so trains may need to circulate slower than usual, which means delivery times get increased, which lowers the country's overall productivity.
If all this happens during a boom cycle, the consequences might be offset by the growing economy: the laid off workers get a job elsewhere, the tracks factory might sell steel bars to someone else so their workers still get a new computer, other people might be willing to pay for more expensive dishes at the restaurant where the railroad workers used to eat, etc.
If this happens during a bust cycle, where the private sector is cutting expenses and thus creating a vicious circle, the State adds to the vicious circle instead of pushing the economy back into growth.
So during a recession you don't cut spending. You either:
Lower the interest rate, so loans for both production and consumption are cheaper
Increase spending by printing money, which causes inflation and might lead to stagflation if it doesn't create growth, which increases povery
Increase spending by issuing debt which can have nefarious long term effects if taken to a extreme: if the debt is in the country's currency, it can lead to inflation as the country prints money to pay it. If it's in a foreign currency, it can lead to a default.
Do nothing, which means you're still in recession