Yes, some of the series have series-contained running jokes. Wikipedia handily documents some of them on the Top Gear series-specific pages. Series 11 has:
The series included running jokes appearing in each episode: Clarkson showing an image he claims to have found on "the Internet" which is censored for the television broadcast; and, in the news section, May appearing to feign enthusiasm whilst presenting a brief news item about the Dacia Sandero with no further discussion before they move on to the next item.
Series 12 has:
The running joke regarding the Dacia Sandero continues from series 11, with the alteration that Clarkson is now the one making the announcement with feigned enthusiasm, to utter indifference from May (instead of vice versa). Each news segment also features one of the hosts wearing something unusual, like slippers or a garishly patterned shirt. The only reference to this is that one of the other hosts will, at some point in the segment, ask if they're wearing it for a bet.
Some of the episode summaries comment on running jokes as well.