The House Administration Committee yesterday began hearings on a proposal to provide for partial public financing of House campaigns, a bill the Democratic leadership has made a priority for this session.

Committee Chairman Frank Thompson (D-N.J.) said a bill would be marked up in early May, after hearing from candidates, party officials and special interest groups.

There have been several attempts in recent years to pass public financing for congressional campaigns, but Thompson said the bill would be successful this time because a "corner was turned in the '78 elections." He said special interest contributions have escalated from $12 million in 1974 to $32 million in 1978 and "we can't permit this escalation to continue unchecked."

But for the first day of the hearings at least, the bill's supporters were on the defensive.

Federal Election Commission Chairman Joan D. Aikens, the first witness, got into a squabble with the bill's supporters by estimating the cost of the bill as between $35 million and $44 million. Supporters of the bill contend that looking at the 1976 House races, the cost would come to about $22 million.

Aikens also contended that the certification process in the bill, to determine whether candidates deserved federal funds to match contributions they raised, was not adequate.

The bill gives a minimal role to the FEC in the certifying process, and requires a 48-hour time period for funds to be certified and paid. Aikens said to certify funds without comprehensive review would "constitute a drastic departure from the type of review that the commission undertakes for presidential primary candidates," and would undermine public confidence in the operation.

Thompson argued, however, that only once case of fraud was found during the 1976 presidential primaries and that was discovered in an audit after the federal funds had been certified.

Rep. Bill Frenzel (R-Minn.) raised another problem with the bill. He noted there would be a limit of $180,000 on the amount of federal money that could be given to candidates in any one congressional district, with each candidate eligible for up to $60,000 in federal matching funds for the general election only Frenzel wondered what would happen if a Republican and two minor-party candidates qualified for their $60,000 before a Democrat did, thereby using up the $180,000 limit in matching funds. Aikens said the Democrat would be out of luck since there was no mechanism of pro-rating or even distributing the money.

The bill affects only House races, but a similar bill affecting Senate races has been introduced by Sen. Edward M. Kennedy (D-Mass.).