To reiterate, the most effective solution is to use Text (vector) and shapes (vector) to recreate the logo from scratch. Keeping it in vector form when you enlarge or reduce before it is eventually rasterized will give you the cleanest edges.
Now, for making the existing logo a bit cleaner.
The PNG file format is nice as it has four channels, RGB and A for alpha or transparency.
Your starting image is shown below, along with the Layer stack.
The first step is to separate out the alpha/transparency channel to a Layer Mask. That is done with the command Layer > Layer Mask > From Transparency. The overall image is the same yet you can see from the Layer thumbnails that they are separate.
To look at the RGB channel by itself, all you do is right-click on the Layer Mask and choose Disable Layer Mask. What I found unusual was all the strange patterns left in the RGB portion of the document.
To view just the Layer Mask or alpha channel of the image, simply use Option + click on the Layer Mask thumbnail (Alt + click on a PC). This is the inverse of the logo. What it means is that everywhere it is white, it is Opaque, and what will show is that portion of the RGB image. For the fully black area, those are 100 % transparent and what will show is what is in the Layer below ths Layer (e.g. white if the background is white)
Notably, the transparency primarily determines the logo, and there is no reason for the RGB channels to be anything but black.
So the first step is to click on the RGB or Image thumbnail in the Layer panel and fill the Image with all Black. This dramatically simplifies how to proceed and correct the image.
Now zooming in to the lower right corner "U" and edge circle, you can notice both the roughness of the edge circle and that the "U" is also quite rough and actually many portions that have no solid black in them and are quite transparent. The following steps will show how to clean this up.
Your right click on the Layer Mask and choose Select and Mask. You will get the panel in the second image below.
You have many options in this panel. In the view option near the top dropdown, I selected "On White." For the contrast sliders, I used 50%, and for the smoothness slider, I chose 50. As you can see, this firmed up the solid portion of the letter and also made the edges quite a bit smoother. There is a tradeoff as higher numbers of smoothness will make some of the sharp points on the text smoother as well. You adjust the sliders to achieve the desired result you want.
Returning to the overall image, the logo remains the same in this backed-up view, and now you can look for other issues that came with the Logo. Since we will be making changes to the Layer Mask only, and not the RGB image (which should remain all black), click again on Opt + click on the Layer Mask to open it for viewing and editing.
When starting with the original image, there was some image "noise" where the large containing circle was broken for the text. With the settings I had, I did not find any to correct, yet you may with a different set of settings with Select and Mask. If you do, for white slivers you would paint with black and if you and any black slivers, you would paint with white.
Hope this helps
John Wheeler