Lee Chappell, Sam Corcoran and Ben Martin all had their contracts terminated last week, while Louie Theophanous joined Chelmsford City.
The quartet have been replaced by players of less experience, but Allinson insists their stays at the club were not cut short for financial reasons.
"I think that is an absolutely disgraceful thing to say. I think it is a poor question," Allinson said.
"The owners have never ever put me under pressure to cut the budget. We have increased the budget in certain areas because we have brought extra physios in.
"We have brought players in from Ipswich, Peterborough and Stevenage. It certainly isn't a budget cut."
On the pitch City's troubles continued on Saturday as they went down to a 3-0 defeat against Ebbsfleet United.
After competing in the early stages at Clarence Park, three goals in the space of eight minutes ended any hopes of City securing an important win.
Sam Deering got the ball rolling in the 27th minute, before Danny Kedwell and Jack Powell completed the scoring.
The defeat means City have just one win in their last 16 games and slip to ninth in the National League South table with eight games to play.
Allinson bemoaned a frantic period, which was riddled with sloppy St Albans defending, and claimed the spell cost his side after an otherwise solid display.
"We put in a good fight for 30-35 minutes, but five minutes of madness cost us," he said.
"We failed to clear our lines on a couple of occasions and we got undone with a soft goal from a throw.
"We were in the game, had a couple of half-chances and the shape was really good."
City's hopes of making the play-offs look increasingly ropey, with a ten-point gap separating them from fifth place Poole Town.
The Saints manager is not giving up on leading his side to promotion and is imploring everybody associated with the club to remain positive.
"I have to be positive in terms of where the players are because the response today was outstanding," he said.
"We will prepare for Gosport and I will expect us to win the game. That is the way I have always been and there is no negativity.
"The whole club has to be positive and take things forward. We ask everybody to come together and we are going through a really difficult period."
Gosport's own dire form - they have taken one point from their last 10 league games - should give City belief of turning a corner on Saturday.
The hosts are five points adrift of safety in 20th position in the table and come into the game on the back of a 2-0 loss to Margate.
Victory for the Saints would see them end a five game winless league run.
There is a St Albans City team and supporters coach going to the Gosport game on Saturday.
This will depart at 10.15am prompt from the Clarence Park entrance to the ground.
Since the coach has a limited number of seats available, bookings should be made as early as possible by calling Ian Rogers on 07714 145862.