You will need to group the object, but the gradient tool for some reason does not allow you to apply a gradient to a grouped object directly (as you can see me demonstrate in the middle of the attached GIF). I'm not sure if this is a bug or not - but there is a workaround!
Group the text and the object, then add a new fill from the appearance panel.
There could be different ways to achieve this, but here's one way:
Convert text to outlines via Type → Create Outlines. Keep an editable copy if you need to.
Select all, including bullseye and apply gradient. This will apply a gradient to each individual letter and shape.
Hit G and drag your mouse around around, see what happens.
Take it from there and ...
Illustrator hasn't an one click transformation which solves your problem. But if you make a high resolution, say 2000 x 2000 pixels or more sized raster image copy of your shape you can make a polar to rectangular distortion filtering in Photoshop (ADD: check, if that is included to Illustrator's own collection "Photoshop's effects")
In Photoshop make an ...
I suspect that you may have applied some graphic styles/effects to the layer. To fix it, try the following:
Select the problematic layer in the layers panel by targeting it.
Open the Appearance panel (Shift+F6) and remove any strokes, fills, or effects in there. The panel should look like this.
Solved: in order to keep the gradient continuity and have the possibility of moving the shapes without changes on their gradient proportion, make a compound path and then apply a pathfinder "Divide". Bingo!
This answer is for Photoshop.
To begin with, you'd really need an undistorted copy of the letterhead. Trying to edit an already distorted image is less than ideal because it will degrade the image quality quite a bit. However it could be done. You could try to use the Perspective Crop tool to undistort it first.
The example below shows use of the ...
Do the following in PS:
Assuming you already selected the particular area using pen tool.
To transform a selected area. Then choose CTRL+T or Edit > Free Transform or Select > Transform Selection
Select any corner point and hold CTRL and move as you need.