Re-arranging data on a worksheet based on contents of Column A
Your nesting ifs have things in common, so you can restructure it
Option Explicit
Public Sub FixAbstracts()
    Dim lastRow As Long
    lastRow = Cells(Rows.Count, 1).End(xlUp).row
    Dim i As Long
    Dim rowCount As Long
    Dim isEmptyCell As Boolean
    
    For i = lastRow To 2 Step -1
        isEmptyCell = IsEmpty(Cells(i, 1))
        If isEmptyCell Then
            rowCount = Application.CountA(Rows(i))
            If rowCount &gt; 0 Then Cells(i, 1).Delete xlshift:=xlToLeft
            If rowCount = 0 Then Rows(i).Delete
        ElseIf Not IsNumeric(Cells(i, 1)) Then
            Cells(i - 1, 13) = Cells(i, 1)
            Cells(i, 1).Delete xlshift:=xlToLeft
        End If
   Next i
End Sub

Now you have a For loop so no need to increment i.
It also steps backwards bottom to top so nothing is skipped. If you delete row(10) and go to the next row, the original row(11) is skipped. That's a result of the For loop, but it prefer it much to a Do While.

Is it empty?
We also declared a boolean to check for empty so it's only done once per loop. Now we base our if on that to reduce multiple checks.
So now the logic is

empty? Yes? Count the row. Now do something
empty? No? is it a number by itself? Yes? Do something

Is it a number?
Let's look at the logic of yours

is it a number? yes? goto next iteration
is it a number? no? is it empty? yes? Do something
is it empty? yes? count the values in the row = 0 then do something
is it empty? yes? count the values in the row &gt; 0 then do something

See the reduction? We get rid of the initial isnumeric because the loop will iterate by itself. We also check for empty only once and we only count once.
Not to mention your code is very difficult to follow.
If Application.WorksheetFunction.IsNumber(Rng.Value) = True Then
    i = i + 1

ElseIf IsNumeric(Rng.Value) = False And Application.WorksheetFunction.CountA(Rng) = 1 Then
    Rng.Copy Destination:=Rng.Offset(-1, 13)
    Rng.Delete Shift:=xlToLeft
ElseIf IsEmpty(Rng) = True And Application.CountA(row) &gt; 0 Then
    Rng.Delete Shift:=xlToLeft
ElseIf IsEmpty(Rng) = True And Application.CountA(row) = 0 Then
    row.Delete

Else: Stop
End If

I had to reread this a bunch of times and I'm still not sure if I'm doing all the checking you are. You aren't consistent with how you're checking for a numeric value, you use a worksheet function and the vba function, there's probably no need to use the worksheet function if your data is standardized.
There's also no need to check if something like isempty = true as the result of isempty is either true or false.