A member of the public contacted the animal welfare charity on Wednesday (11) after witnessing the bird of prey fly out of woods near to Darnholme Crescent before crashing into a car.
The bird was found lying on its back with its wings spread out.
RSPCA inspector Claire Mitchell, who is investigating, said: “This poor sparrowhawk was in a very bad way after being shot and after further assessment, very sadly the decision was made to put the bird to sleep to end its suffering.
“The injury it sustained to its wing and chest were so serious that rehabilitating it would not have been possible.
“It is likely that the bird was shot with an air gun while in flight, and it is very concerning that somebody in the area was taking shots at wildlife in this way.
“Whilst there are some shooting practices which are legal, it is an offence under the Wildlife & Countryside Act to intentionally injure, kill or take a wild bird, except under licence.
“Anyone found guilty could face an unlimited fine and/or six months imprisonment.
“I am urging anyone with information about this incident to contact us on 0300 123 8018.”
The RSPCA received 231 calls about air guns and other weapons across England and Wales between March 24 and the start of October, during lockdown and the weeks that followed as restrictions eased.
Claire said: “Our experience tells us that the main animals that people take pot shots at are cats and wildlife. They are normally the animals that are more susceptible to these incidents simply because they are out in the open with no one to protect them.
“The injuries caused by these sorts of weapons are horrific and often fatal.”
She continued: “These figures will only show part of the picture.
“Wild animals that are harmed in this way are not always found, so are not reported to us.
“When cats are harmed in this way, hopefully they are taken straight to the vet - and not all of these cases are reported to us, either.
“In any case, the devastation these weapons can cause an animal are terrible, with many being maimed, scarred for life and even killed.”
The RSPCA is calling for tighter controls on air weapons.
They say this, along with better education and explanation of the law when buying an air gun, and requirements that everyone must receive basic safety training before being allowed to walk out of the shop, could reduce the number of animals harmed in this way.
The RSPCA is continuing its vital work rescuing animals during the Coronavirus crisis.
To help the RSPCA keep rescuing animals and keep their animal hospitals and centres running for emergency treatment and round the clock care through these unprecedented times, your are invited to donate whatever you can spare at www.rspca.org.uk/covid .