The work will help reduce flooding risk to 238 properties in the city centre caused by such abandoned objects, the Environment Agency said.
It coincides with the Destructor Bridge on the river being removed as part of a new development.
On Monday, shopping trolleys were pulled out of the river by the agency.
Matthew Symonds from the trust said: "Navigation can be tricky along the River Avon and no wonder given the amount of stuff that's down there."
The stretch between Windsor and Victoria bridges is being focused on after the agency carried out a survey of flood defences.
An agency spokesman said: "Large objects such as cars and shopping trolleys can cause serious problems when abandoned in a river.
"They can increase flood risk by disrupting flows and cause a hazard to navigation."
The same contractor that is removing the Destructor Bridge will carry out the Â£20,000 river clean up, which is being paid for by the agency.
The Canal and River Trust has appealed for people to stop dumping objects into the river due to the navigational hazards and the impact on wildlife.
Work is expected to be completed by the end of the week.

The Northern Irishman carded a third-round 68 for a 16-under-par total, with American Rickie Fowler his nearest contender on 10 under.
McIlroy, 25, won from seven back at the PGA Championship at Wentworth in May.
"I know leads can go quickly," he said. "That's why you can't let yourself think about winning."
McIlroy was tied with Fowler on 12 under after 13 holes but eagles at the 16th and 18th helped him to the biggest 54-hole advantage at the Open since Tiger Woods led by six at St Andrews in 2000.
But McIlroy - whose father Gerry and three friends placed a bet £400 at 500-1 10 years ago that he would win the Open by the time he was 25 - is well aware what the pressure of leading a major tournament can do to a player, having blown a four-shot lead on the final day of the Masters in 2011 and eventually carded an 80.
Media playback is not supported on this device
"I've been on the right side of it and I've been on the wrong side of it," said the 25-year-old, who has led from the opening day.
"You've just got to stay completely in the moment and that's what I'm going to do for all 18 holes tomorrow."
Fowler will tee off with his old rival in the final pairing on Sunday, having carded a third-round 68, and has not given up hope of overhauling him.
"If I can get off to a good start and put pressure on, there is a lot of golf to be played," said the 25-year-old American, whose rivalry with McIlroy dates back to when they faced each other in the 2007 Walker Cup.
"Being alongside Rory is an advantage. We are going to have a good time, so much can happen so quickly."
Fowler drew level at one stage on Saturday but stumbled down the back nine before also carding a 68.
Sergio Garcia is one shot behind Fowler and level with American Dustin Johnson but the Spaniard also held out hope of landing his first major title.
Media playback is not supported on this device
"I am a little bit disappointed with the way I played the last two holes," said Garcia, whose third-round 69 would have been better but for a bogey at the 17th and a scratchy par at the par-five 18th.
"It is never easy being in the position of leader. Rory is a great player but you never know, anything can happen.
"But you can't just attack everything because you are not going to hit good shots. You have to attack where you feel comfortable attacking."
Meanwhile, world number one Adam Scott has not ruled out the possibility of somebody setting a record score for a major championship on the final day.
The Australian carded a third-round 69 to finish 10 shots behind McIlroy but believes a score of 62 is achievable, if Sunday delivers good weather.
"If we have similar conditions to today, there are low scores out there," said former Masters champion Scott.
"If you can get it going for 18 holes, there's a chance for that major scoring record to be broken.
"I think out of the group of the guys up there, there's probably a low score in one of them."
Cars and mopeds are expected to be lifted from the River Avon in Bath during the first dredging operation for 20 years.
[NEXT_CONCEPT]
Rory McIlroy is taking nothing for granted despite carrying a six-shot lead into the final round of the Open at Royal Liverpool.