The Reagan administration has approved a second round of budget cuts, totaling about $10 billion, which strike at a host of politically sensitive areas, including veterans benefits, child nutrition, job training for the poor and minorities, and water projects, White House sources said yesterday.

The cuts, approved at a Cabinet meeting Friday, included about $6.5 billion covered in broad form in Preidient Reagan's budget address Feb. 18, and about $3.5 billion made necessary by the discovery last week that the administration had underestimated federal spending by $3.1 billion.

The reductions in the first category had been agreed to earlier by the departments, but had not been formally approved by the president or included in the budget message because they were relatively minor ones striking at scores of programs at the agency level, a source said.

These cuts posed no great surprise for the Cabinet officers, a source said, adding that the meeting "was more a matter of breifing rather than a decision session."

The $3.5 billion in reductions, however, is directed at politically sensitive areas, and may add opponents to the collection of interest groups already arrayed against various elements of Reagan's economic recovery plan.

The cuts include reductions for veterans programs, farm price supports, job training, child nutrition, and low-income fuel assitance programs, among others.

The $10 billion in reductions, some of which are still in flux, are directed at the following departments and agencies, sources said:

$800 million in cuts for the Veterans Administration. These include minor programs run by the VA , a construction slowdown, the cancellation of a number of hospital projects, and elimination of flight training as a category of education eligible for GI benefits, a perennial target of budget cutters.

Also proposed are an increased interest rate on VA insurance policy loans -- it's now 5 percent -- and a change in the way diability benefits are calculated for retired veterans. No change, however, is contemplated in basic education, retirement or health benefits.

$900 million in cuts at the Labor Department. The biggest item here is the consolidation of various manpower training categorical grants programs into block grants for states and localites. Budget cutters estimated this would save $250 million.

These programs include "national emphasis" projects designed to aid specific groups like Indians and other minorites. But the Job Corps, originally slated for a major reduction, has been largely spared after pleas from Congress. Reagan has already announced the elimination of public service jobs, and the Labor Department has ordered a freeze on such hiring, effective Monday.

$2 billion in cuts at the Department of Health and Human Services. Among these, the administration wants to combine a $1.8 billion low-income fuel assistance program with 39 other federal health and social services programs into one grant for the states in 1982. Total spending for the 40 programs would be reduced to 75 percent of 1981 levels.

$1 billion in cuts at the Agriculture Department. These are said to involve Farmers Home Administration loans, school milk programs and farm price and farmer income support programs.

Also included in the Agriculture Depatment trims, according to congressional and White House sources, is a reduction of $250 million, or about 25 percent, in spending for the supplemental nutrition feeding program for low-income pregnant women and infants. The program, called WIC (women, infants and children), began as a relatively modest $50 million effort in 1975, but the ballooned to a $1 billion programn in Carter's last budget.

$463 million in cuts in water projects, mostly those of the Army Corps of Engineers, Sources said these cuts would be concentrated in the middle of the country, and some of them are ones that President Carter tried unsuccessfully to make.

Reagan, in his original message. proposed some relatively minor water projects cuts, not large enough to create much political resistance. But some of these new ones evidently will go further. Carter, early in his administration, proposed a water project "hit list" that met with enormous resistance on Capitol Hill and helped sour his relationship with Congress for much of his first year in office.

In his speech to Congress, Reagan urged budget cuts totaling $41.4 billion. He identified 83 specific program cuts totaling $34.7 billion, and promised to detail additonal cutbacks when a completed budget is submitted March 10. These now approved cuts range from about $10 million to $30 million per agency, sources said.

Last week, the administration admitted it had miscalculated federal spending by between $3 billion and $6 billion, making further cuts necessary to reconcile the budget. White House press secretary James S. Brady yesterday attempted to pinpoint the spending underestimate at $3.1 billion.