Praise be to Allah, and peace and blessings be upon the Messenger of Allah and his family.
The ruling on putting up an awning is similar to what the jurists have discussed about putting up a balcony. They stated that it is not permissible to put up a balcony if it will harm or bother passers-by or neighbours.
It says in Fatawa wa Rasa’il Samahat ash-Shaykh Muhammad ibn Ibrahim Al ash-Shaykh, 7/245: It is not allowed to build a balcony that protrudes from the building on wooden beams, whether it is small or large.
The three imams [Malik, Abu Hanifah and ash-Shafa`i] favoured the view that it is permissible to do that, so long as it will not cause any harm to passers-by or neighbours, and this is more appropriate. Similar to wood are the long metal beams on which wood is placed. In fact this refers to a balcony and other similar things that could block the air of the street. End quote.
In al-Bayan fi Madhhab al-Imam ash-Shafa`i, 6/252, it says: If a balcony protrudes into an open ended street (that is, not a cul-de-sac), we think that if it does not bother the Muslims, it is permissible and should not be prevented. This was stated by Malik, al-Awza`i, Ahmad, Is-haq, Abu Yusuf and Muhammad (may Allah have mercy on them). End quote.
Shaykh Ibn `Uthaymin (may Allah have mercy on him) said in his ruling: The correct view is that there is nothing wrong with attaching something ordinary to a building that does not bother people, with the permission of the authorities. But if it is something that does bother people, then it is not permissible, even if the authorities in the city, such as the mayor, gave permission for that.
End quote from ash-Sharh al-Mumti` ‘ala Zad al-Mustaqni`, 9/254.
People usually install awnings for protection from the sun and rain, to conceal the occupants of the house, and to ward off annoyance that can come from upper floors – as mentioned in the question. So what appears to be the case is that it is permissible to install this awning, but you should adjust it when installing it so that there will be room for the neighbours to lower their buckets, because they are clearly bothered by that. But at the same time, they should put a stop to the harm that is coming from their side. If they do not do that, there is nothing wrong with you leaving the awning, because you did not install it for the purpose of annoying and harming your neighbours; rather you did it so as to ward off the annoyance caused by them.
And Allah knows best.