My inner kyu level player says Black should kick at a.
The book by Rob van Zeijst has this solution diagram.
Seems to me that the corners/side is still invadable in either diagram. The kick (a in the first diagram) is often used to make the opponents stones heavy. With the marked Black stone in place, this seems particularly effective, White is denied the common extension to in the following diagram. So why not kick?
Andy: Indeed. What is white trying to accomplish with here? Presumably and are attackable due to considerations of the board not shown in the diagram and the white stones have sufficient support from the center such that their life is not a concern. Then white can use them in support of an invasion under the black stones on the left side. Otherwise, this doesn't seem playable for white and is just wishful thinking on black's part. If and are not attackable, then it isn't much of a loss for white to let them connect to black's stones on the left side so white might as well enter the corner directly rather than connecting the peep.
Dieter: Let's have a look at the corner invasion as an answer to Black's peep. I assume this is best for both. Since is labeled as a genuine threat, must be big. In any case, Black's connection is strong and big.
I think that here, White can play elsewhere or even check Black's marked stone with . The threat of is not so big.