4-4 point, double low approach, outside diagonal

Diagonal answer  

Charles Matthews This diagonal play has experienced a revival in pro games, associated with new thinking about central influence and possible tenuki variations.

After here, at the 3-3 point is normal; and now Black plays a, b, or tenuki allowing White at a.

The problem with this joseki for a weaker player is that Black at a can leave a black wall, but without a pincer attacking and also with some potential shape problems. Clearly this isn't troubling at pro level, but amateurs can come to grief here.

Black's answer a  

After Black's main options are at e, f and g. Playing g, which is a move useful in handicap games when Black has a stone already at the circled point to attack , isn't very forceful here. Black at e is only seen in old games.

Black cannot tenuki  

After , black cannot tenuki. White's shape is too good after , . If black wants to tenuki, it should be before , not after.

Black presses  

Probably is the best idea here, but the bulge play gives Black some shape difficulties.

Black presses  

In games of Hane Yasumasa based on the Chinese opening , , was used to build influence for Black on a large scale, treating the cutting point at a as secondary.

White's mistake  

This would be a big mistake as it doesn't take advantage of Black's shape problems.

Joseki  

When is present, this is played in many pro games.

Huh?  

I think the correct response to is a. But not knowing what is suppose to do, I have no clue how to develop my corner group. (My guess is that helps B runs out to the center, but is that it?)

20k guess: leaves two possibilities - live in the corner, bigger than if hadn't been played (eg with 3-3), or run into the center (eg with b). In a nine handicap game, black probably isn't clever enough to tell which is better. Therefore, treat them as miai and play somewhere else.