Last week 85,000 households in Hinds and Adams counties in Mississippi received a friendly letter from President Reagan telling them about his friend, Liles Williams.

After praising Williams' virtues as a civic leader and his devotion to the Reagan economic program, the president concluded his letter by saying, "My administration needs his help in Congress."

That letter, paid for by the Republican National Committee, is among the many reasons that Williams, vice president of a Jackson, Miss., electrical supply firm, is widely considered the front-runner in a special election today in Mississippi's 4th Congressional District.

But even the support of Reagan, who is highly popular in the conservative district, is not likely to be enough for Williams to avoid a runoff election July 7 against one of the four Democrats in the race -- probably Jackson attorney Britt Singletary.

Under Mississippi election laws, if no candidate wins a majority of the vote today, the two top vote-getters, regardless of party affiliation, face each other in a runoff.

The special election was made necessary by the resignation in March of Republican congressman Jon Hinson after his arrest on a morals charge. For most of the campaign, nothing was said of Hinson and the homosexual incident that led to his resignation.

Then, last Friday, Singletary called a news conference to charge that the Republican power brokers who had backed Hinson, and thus had "embarrassed" the district, had handpicked Williams to be Hinson's successor. He also predicted that the response from the Williams camp to his charge would be to portray him as a "desperate politician."

He was correct. "The people of this district are not going to buy guilt by association," said Norman Turnette, Williams' campaign manager, who accused Singletary of panic in the face of gains by another Democrat, McComb Mayor Wayne Dowdy.

Williams enjoys many advantages, including a campaign budget of $220,000, with probably at least that much also available for the runoff.

Williams has disassociated himself from Reagan's proposed Social Security cuts, but otherwise has wrapped himself in the president's economic program. In a likely preview of the runoff campaign, the Williams camp recently has run ads linking all Democrats to House Speaker "Tip O'Neill and his liberal friends" and urging Mississippians to "Send a Message to Tip O'Neill."

Williams also opposes extension of the Voting Rights Act, a popular position among the district's majority white voters.

Dowdy has gone all out, largely with his own money, for the black vote. Campaign reports show that as of early June the wealthy, first-term mayor had spent $103,000, including $78,000 of his own money.

Ellis Woodward, a Singletary aide, concedes that Dowdy, who has the AFL-CIO's endorsement, had made gains but says he also "completely alienated his white base" and could finish behind another Democratic candidate, State Sen. Ed Ellington.

The Singletary camp is so confident, Woodward said, that it has anounced that Mississippi's popular Democratic governer, William Winter, will take to the road Thursday with Singletary for the runoff campaign.