Australia's Hawke: Can he cozy up to US and still satisfy his left wing?

For many outside of Australia, that country's image in recent years has been polished by exports of good movies, thumping rock music, and that household staple, the vegetable spread Australians affectionately call ''vegemite.''

This week's visit to Washington by Australia's Labor Prime Minister Robert Hawke is a reminder that Australia also has another goal - to combine its American ties with the broader role it seeks to play in Asia.

Since Mr. Hawke became prime minister in March, Australia has modified some previous Australian policies of former Liberal Prime Minister Malcolm Fraser. The new government has sometimes broken with its own party's policies to reach out to Indonesia, Japan, China, and Vietnam with initiatives that seem aimed at making this island continent a listened-to mediator in the affairs of Asia.

In Washington, the focus has been on the way in which the new Labor prime minister handles a tricky challenge more directly related to the US. This challenge: Can Hawke maintain a good working relationship with the US and still retain the support of his party's left wing? The left thinks Australia-US military cooperation has gone too far.

These critics have never been happy with the stationing of US personnel at communications and satellite installations at North West Cape, Pine Gap, and Nurrungar. The Labor Party has lived with them, saying they are acceptable insofar as they are an early warning system to deter a Soviet attack. Since 1974 , Australian governments have had to convince their critics that US personnel, including the CIA, were not secretly misusing these facilities in a way that would make Australia a priority target for Soviet nuclear attack.

Hawke has bowed only slightly to left-wing pressure by suggesting an update of the 1971 ANZUS pact binding Australia, New Zealand, and the US in a treaty of defense. In public the prime minister has in no way challenged the base system beyond suggesting slight modifications such as stationing of an Australian military officer at the Pentagon to improve intelligence cooperation between the two countries.

One sign came in Hawke's Washington news conference June 14 after a two-hour meeting with President Reagan, and a meeting with Defense Secretary Caspar Weinberger.

Hawke declared he had established a warm personal relationship with Reagan while airing some differences. He mentioned his request for a softening of a US ban on Australian purchases of some sensitive US military technology, and Australian concern that subsidized US farm products might compete unfairly with Australian produce.

But Hawke's prospects for getting stronger intelligence cooperation was dealt a blow just weeks before his visit to the US, when the Australian newspaper the National Times printed sensitive information on US-Australian intelligence cooperation. Court action limited future revelations, but concern among US intelligence agencies grew.

Another problem for Hawke is President Reagan's policy of building up US military power against the Soviet Union. This makes it more difficult for the prime minister to pacify his left wing, some of whom see US militarism as increasing the likelihood that US bases on Australian soil may became Soviet targets and draw Australia into war.

The Australian press has suggested that President Reagan's long-term policy of preparing for a ''star wars'' defense in outer space may require that US bases in Australia be used not only for early warning of Soviet attack, but to identify and plot Soviet missile targets during a war. This would not be compatible with Labor policy on the bases.

Hawke reached Washington after stops in Papua-New Guinea, Indonesia, England, Paris, and Geneva, and following a number of initiatives demonstrating Australian involvement in Asia. After receiving a Chinese delegation headed by Prime Minister Zhao Ziyang in Australia, Hawke pledged he would lend his good offices to help smooth Chinese-American frictions.

Hawke also has stressed ties with Japan. Diplomatic and economic relations with the Soviets have been normalized.

The government announced that in accord with Labor Party policy it would resume humanitarian aid to Vietnam. But recently this has been delayed so as not to upset negotiations between Vietnam and the Association of Southeast Asian Nations. Australian spokesmen have expressed cautious hope that Australia can help to create a compromise by maintaining good relations with all.

Most dramatically, Hawke has suggested that his Labor Party may have to change its policy of barring military aid to Indonesia until it allows self-determination for the former Portuguese colony of East Timor, annexed by Indonesia in l976. This apparent effort to improve long strained relations with Indonesia appears to be further recognition that Australia's future requires close ties with Southeast Asia.