Hmmm...both are fine, but I would tend to say #2 before I would say #1, just because of personal preference.

When I think of the phrase "for THE second time", I feel like it should have more to it. It should have the phrase "this month" or "this week" or "today" at the end of it to make it more of a complete thought. Like, "He needs to wash his car for the second time this week."

'a' vs. 'the' with ordinals

When you do something a second, a third, a fourth time, ..., you typically have not planned to do it a second, a third, a fourth time, .... Circumstances just worked out that way.

When you do something the second, the third, the fourth time, ..., you probably planned it that way, OR it is the second, the third, the fourth time,... in a longer series of times.

Compare:

The walls were a dark green, and we preferred white. We gave the walls a coat of paint. It didn't cover the green well enough, so we gave the walls a second coat of paint. (the not possible here. We hadn't even considered a second coat at first.) Even the second coat didn't cover the green. We applied a third coat.

The walls were a dark green, and we preferred white. We were concerned about how well white paint would cover the green walls. We consulted a few painters and found that it would probably take a minimum of three coats of paint to do the job right. So we bought enough paint for three coats of paint. We applied the first coat Saturday and the second coat Thursday. Next week we plan to put the third coat on. Then we'll see if it's OK. We're hoping we won't need a fourth coat!

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In summary, both are correct in isolation. You simply need to choose the one which is correct for the context.