I had cardinals do that last spring. I stripped the glass window with painter's tape. It worked for a few weeks but they returned.

Tape a large picture of a hawk to the glass.

I concur...the white feathers on the tail is the give away

however at first glance at the 1st picture the Titmouse immediately came to mind

The Carolina Wren will also demonstrate the same behavior of resting by latching onto a window screen and flying off/on again....they're not trying to get in, they've just found a 'perch'

like the Blue-Gray Gnatcatcher (which is the bird you have) the Carolina Wren is a year round resident (but the Wren would never be confused with the B-G Gnatcatcher because of the different coloring)

As to why the bird is doing this, did someone close to you recently pass? Sometimes spirit returns In this gentle way to get your attention.

There you go, that explains it <>

I agree with C. Breit. It is a Blue-Gray Gnatcatcher which looks like a mockingbird but is 3-4 inches smaller. Also there is no tuft.
Happy Birding!!

He used to live there before you moved there. LET HIM IN!

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I concur with the blue-grey gnatcatcher.

First I have to say, what a cutie. Everyone had good possible explanations, but (now don't laugh at me) what came to mind was maybe it's a loved one who passed. Ever time I and my siblings see a fruit fly, we say it's my dad. Too many stories and coincidences. LOL

Well, Google said the mockingbird IS the Florida state bird.

It is a blue gray gnatcatcher, white on tail feathers and eye ring. It is catching gnats 😆

I hang suncatchers in the windows. The birds don’t seem to like them.

This has happened to me plenty of times and was told it sees itself and during mating season thinks it is a mate. It happened to me with a woodpecker and a cardinal. Sometimes it last for a couple of days. They come certain times of the day and just peck on the window.