How much additional mass does the Earth gain each day from solar radiation?

There's an answer to your question, but it's not all that meaningful.

The sun strikes the Earth with $1.5\cdot10^{22}J$ of energy every day. Using $m=\frac{E}{c^2}$ we find this has a mass equivalent of 166897kg.

However, the Earth does not actually gain mass this way. The Earth is also radiating energy into space, continuously. If we assume the average , so the amount of energy coming into the system equals the amount of energy leaving the system. As a result, the earth is not gaining mass by this at all. (or if any, its a small amount attributable to global warming).

We also gain about 40000kg of space dust every day, and lose about 95000kg of hydrogen from the atmosphere. You win some, you lose some.

Not only does Earth not retain any significant amount of energy it gets from the Sun, but it actually loses its thermal energy (by radiating more than it is absorbing) while its core is cooling down. Core cooldown rate is estimated at about 50 terawatt, which translates to about 2 kg of mass lost per day.

As Cort's answer explains, there are other sources of mass gain / loss which are several orders of magnitude more significant than this mass loss due to radiation.