Oh dear, Your method isn't the best
For fanlights that are leaded it is far, far easier to use a hand sprayer to lightly mist the glass and then buff clean with a dry scrim, or better still a large (dry!) microfibre cloth.
As soon as your scrim or cloth gets too damp to be effective, change it for a dry one, but make sure it is a clean one and not just one you have let dry out.
Washing and rinsing out your scrim in your bucket is ok if all you are using it for is to clean the frames and sills and mopping up after you have squeegeed off the window, and if you keep washing and wringing it out you can get away with cleaning the fanlights with it, but while the glass is still damp you HAVE to buff with a clean and dry microfibre or scrim.
If your scrim is full of holes then you've worn it out, you should have a dozen or more of them at hand so that you can rotate them through the day.
When I was full time trad I could make do with 3 or 4 scrim a day.
I'd start the day with them clean and dry.
As soon as the first one got too damp to be able to detail or buff the glass dry with it became my wet scrim for the day.
The others I would rotate through the day.
The following day you start with a fresh set of clean, dry scrims.
A dozen scrim would take months and months before any holes begin to show in them.
Though were I trad only now I would have about a dozen large mocrofibre cloths and a half dozen scrim.
The scrim I would use for mopping up and wet work, the microfibre for leaded windows, georgian windows and detailing and any other glass that I wouldn't use a squeegee on.
Dump the tea towels too, it looks so unprofessional, and not remotely close to being halfway as effective as the proper equipment.
One of the best (and cheapest) window cleaning sprays to use is Tesco's own brand solution, 80p or so for 500ml. It really is very effective.
Ian