To gain $8\%$ after allowing a discount of $10\%$, by what percent must the marked price be hiked from the cost price?

Let the Cost Price ($\text{CP}$) be $100$.
To get a gain of $8\%$, the Selling Price ($\text{SP}$) must be:

$$\text{SP}=100+8=108$$

A discount of $10\%$ is given on the Marked Price ($\text{MP}$), which means:

$$\text{SP}=90\%\text{ of }\text{MP}$$

$$108=\frac{90}{100}\times \text{MP}$$

$$\text{MP}=\frac{108\times 100}{90}=120$$

The percentage hike from Cost Price to Marked Price is:

$$\text{Percentage Hike}=\frac{\text{MP}-\text{CP}}{\text{CP}}\times 100=\frac{120-100}{100}\times 100=20\%$$