Problems with Offset Path

Hannah,

I agree with Theresa that the letters must by wee, and it will be better to work at a much larger size.

Apart from that, the offset is far too large to avoid the big blob.

To look like the image with the desired appearance, the offset must be about 1/40 of the (approximate) letter height.

However, as I (mis)understand it, offsetting is never going to work as desired because it will give unwanted cutouts: the bar of the A will also make apparent cutouts from the stem of the R, and the (right) stroke of the A will give a cutout from the leg of the R.

If you wish to only have apparent cutout from the stem of the R, you can do it as follows, with the A above the R (in the stacking order, as can be seen in the expanded Layer in the Layers panel), Smart Guides being your friends helping you to snap, images shown with a fair font at 72pt on a yellow background, and a Stroke Weight of 2pt.

You can use Helpx search to see more about the terms and ways at the bottom.

0) Select the A and move it above the R in the expanded Layer in the Layers panel, then switch from black fill/nostroke to nofill/red stroke with a Stroke Weight that is twice the desired cutout, this will also give an apparent cutout from the stroke of the A but only temporarily; 2pt Stroke Weight shown in the images, the red stroke colour is chosen so you can see what you are doing;

1) With the Line Segment Tool ClickDrag from the top right corner Anchor Point of the (right) stroke of the A (Smart Guides say anchor when you are within snapping distance) and along the right side of the stroke down past the stem of the R (Smart Guides say path when you are within snapping distance) and let go, then ;

2) Repeat 1) for the left side of the, only between two points on the left side on either side of the stem of the R (Smart Guides say path when you are within snapping distance);

3) Group the two red stroked paths from 1) and 2) and drag the Group down in the Layers panel between the A and the R;

4) Switch from red to white stroke colour, then ShiftClick the R, then in the Transparency panel click Make (Opacity) Mask with Invert Mask ticked.

This ought to knockout/hide the desired parts of the stem of the R.

Click/RightClick to see in full, Click again to get closer, Click Here to see in Chrome 

Edit: Addition of the suggestion I had in mind when I had to leave for a while.

Edit edit: Elaboration of steps and addition of images and Helpx search.

You can use Helpx search to see more about the terms and ways here:

This link is about the Layers panel,

https://helpx.adobe.com/search-results.html?q=layers+panel+illustrator&scope=%5B%22helpx%22%5D&subscope=%5B%5D&limit=10&start_index=0&sort_orderby=relevancy&sort_order=desc&post_facet_filters=%7B%22applicable_products%22%3A%5B%5D%7D

and the first entry is this,

https://helpx.adobe.com/illustrator/using/layers.html

This link is about the Line Segment Tool,

https://helpx.adobe.com/search-results.html?q=line+segment+tool+illustrator&scope=%5B%22helpx%22%5D&subscope=%5B%5D&limit=10&start_index=0&sort_orderby=relevancy&sort_order=desc&post_facet_filters=%7B%22applicable_products%22%3A%5B%5D%7D

and the first entry is this,

https://helpx.adobe.com/illustrator/using/drawing-simple-lines-shapes.html

Seperate each letter in a layer and offset individually. 

If a 4px offset is that large your artboard and text is probably very small. What happens if you scale the text up to a much larger size, then try a 4px offset?