Greetings from Fort Worth, TX, Stephanine !!! I cannot ID Dan Mulharen, but can provide you with the history of the 34trh. Infantry Division, as well as it's combat history for the period you are interested in. The 34th. included three infantry regiments, 133rd, 135th and 168th. They worked closely together and often interchanged positions in the line. Also included were four field artillery battalions, the 125th, 151st,175th and 185th. The 34th. landed in North Africa 3Jan43; landed in Italy 21Sep43; landed across Anzio 25Mar44; participated in these campaigns: Tunisia, Naples-Foggia, Rome-Arno, North Apennines and Po River.
A brief portion of the division's Combat Narrative for the period of your immediate interest follows:
"On 25Mar44 the division landed at the Anzio parimeter and maintained defensive positions when the 133 Inf. Reg. led its attack towards M. Arrestino. Dan probably died on the assault on M. Arrestino
The 133rd. overran Tarquina on Highway 1 on 9Jun44. "
Please note: there is more narrative available, nbut it is outside your focal time of 1Jun44.
Finally, the division was under the VI Corps at this time, MG Chas.W. Ryder,commanding.
Casualties: Killed in Action 2,866 Wounder in Action 11,545 Died of Wounds 484.
My reference for the above: "World War II Order of Battle", by Shelby L. Stanton, published by Galahad Books, NYNY, and well worth the price (About $25 at Barnes & Noble).
I have another excellent reference, and will let you know if I can get any additional information of interest to you.
Me? I was also in Italy in the 985th. Field Artillery Battalion, II Corps, Fifth Army. I too fought the North Apennines and Po River campaigns.
I hope that the above is of interest and benefit to you and your family.