No, not really sure why you would like to add polybrene. We have injected concentrated supernatant just fine, although I think, more regularly, we have injected in lenti which has been centrifuged then resuspended in PBS, or DMEM, or Optimem. Which organ & how do you plan to inject?

I want to inject the kidney with 10 ul of lenti bought from sigma. That means I can just directly inject the viral solution into the organ, right?

You will need to dilute your virus stock to whatever pfu you plan to use. 10 µL is too small to accurately inject with most syringes unless you prepare a larger volume that will remain in the syringe after injection (i.e. load 30 µL for each injection but still only inject 10 µL).

Regarding the injection method, there are several approaches published for viral infection of kidneys, such as retrograde infusion into the ureter, injection into the renal vein or artery, or direct injection into the renal parenchyma. Each of these methods has advantages, particularly in regards to what region of the kidney is transduced. These papers review some of these methodsArticle Lentiviral Gene Transduction of Kidney

Article Comparison of Gene Delivery to the Kidney by Adenovirus, Ade...

Polybrene (or any other ex vivo transduction enhancer) isn’t necessarily going to help in vivo delivery. It can help with monolayer transduction in vitro, but the challenges of delivering vectors to organs is slightly different. As JHL says, route of delivery and titre/dose per kg are the things to focus on.