To determine how many more palms the palm reader must read to reach her goal of $2240, we start by calculating how much she has already earned from the 4 palms she has read.
The earnings from reading 4 palms are:
\[ 4 \text{ palms} \times 80 \text{ dollars/palm} = 320 \text{ dollars} \]
Next, we subtract this amount from her goal to find out how much more she needs to earn:
\[ 2240 \text{ dollars} - 320 \text{ dollars} = 1920 \text{ dollars} \]
Now, we need to determine how many more palms she needs to read to earn the remaining $1920. To find this, we divide the remaining amount by the fee per palm:
\[ \frac{1920 \text{ dollars}}{80 \text{ dollars/palm}} = 24 \text{ palms} \]
Therefore, she needs to read 24 more palms to reach her goal of $2240 this month.
Thus, the answer is 24.