This game was played in June & it was played before the switch to summer rugby in 1996

When did the winter rugby season end then?
The month I mean..

The Dons kicked off the 1996 season against Barrow Braves away in the challenge cup on January 14th 1996 & played them again at home on the first second division league fixture of the new season on 31st March 1996

The first offical Super League summer time game was on March 29th 1996
Paris Saint -Germain v Sheffield Eagles

Lets recap what you have found out from the clues I have given you

The Dons played a game in the month of June at Stanley Park Oval Ground Blackpool
This wasn't a friendly or seven a side game & the team they played is still in existence

Who were the Dons opponents at this match?

You have wrongly posted
Hull KR
Barrow
Sheffield
York

The correct answer can be found within one of the remaining 32 current teams still in existence

Well no point in guessing ...how about another clue?

Droplets of water

Dewsbury (I don't take any pride in getting this).

Well Done it was Dewsbury

Now for part two of the question.

Why was the game being played midweek in June at Stanley Park

A 7s Competition ?

... and my heart sank

No not a 7's competition

Think of something similar to the Magic Weekend or the Summer Bash

I don't think anyone is going to come up with the correct answer so I will put you out of your misery.

In the summer of 1952 there was a Rugby League competition held at the Oval Stanley Park Blackpool it was called the Blackpool Sports Holiday Rugby League Competition.
The Dons played Dewsbury on Wednesday June 25th

Doncaster Team
Griffiths,Cooper,Price,Williams,Norbury,Clough,Heritage,Gronow,Doyle,Belshaw,Guest,Davies,Street.
Referee Mr G.S.Phillips (Widnes)

A bit easier question

When the Dons finished there tenancy at the York Rd Greyhound Stadium they relocated to the Bentley Rd ground can you name the Dons opponents at the first game played at Bentley Rd