This question is unlikely to get the sort of answer we usually expect from this subreddit, because the honest answer is, “not really.” However, the honest answer isn’t a flat “no,” so here’s what’s here:

In 1945, the Fourth Circuit heard Saunders v. Wilkins:

https://casetext.com/case/saunders-v-wilkins

It breaks down like this: Virginia, at the time, had a poll tax. Poll taxes are now held to be a clear violation of the 14th, but all the jurisprudence of the day held otherwise.

Saunders filed with the Virginia Secretary of State to run for Congress as an at-large candidate to represent Virginia. His reasoning was that Virginia’s poll tax disenfranchised about 60% of the state’s voters, and so VA was only entitled to 4 representatives, rather than 9, owing to that clause of the 14th.

Consequently, he argued, since Congress had incorrectly awarded VA with the 9 representatives the census afforded, irrespective of the poll tax, federal law applied which required that any excess representatives beyond the constitutionally appropriate number must be elected at large by the whole state.

He further argued that, even if the poll tax were not unconstitutional in and of itself, the effect still triggered the 14th.

The court found against him on both counts. They said it was a political question, in that the effect of the lawsuit was primarily to abolish the poll tax, or else to effect a change in the state’s electoral process, which was within the state’s sovereign rights to dictate.

Essentially, they concluded that the whole thing was Congress’ job, and, in the meantime, everything squared.

As far as I know, that’s the closest anybody has come to actually implementing those provisions.

Interesting question! Unfortunately, it has a relatively short answer: no.

The Fourteenth Amendment was partly meant to enfranchise African Americans for reasons of genuine Republican belief. But it was also meant to ensure Republican domination of American politics. They felt (correctly as it turns out) that disenfranchisement such as Jim Crow was likely and African Americans overwhelmingly voted Republican. The Republicans also disenfranchised people but mostly on the basis of having participated in the Confederacy. This was protected in the amendment along with the longstanding common law practice of reducing (or even eliminating) the rights of convicted felons.

The second section (the one you reference) was meant to decrease the representatives of the Southern states, especially under the Democrats. It was also part of a long, largely unsuccessful tradition of anti-slavery activists trying to incentivize their position Constitutionally. The 3/5ths Compromise was not meant as some grand statement about the inherent worth of African Americans. It was meant to incentivize the freeing of slaves: a state could increase its representation by freeing slaves. In fact, the pro-slavery delegates wanted them fully counted and the anti-slavery delegates wanted them to not be counted at all. (They wouldn't get to vote of course but it'd increase the representation of the slave states.)

Anyway, the Republicans fully intended to use it. They tried to from the 1870 election until roughly the Compromise of 1877. But they were never able to find enough disenfranchised voters to count. Largely because Jim Crow wasn't really effective until Democratic domination in the South. Even where the Democrats succeeded in gaining control, organized and violent Republican resistance backed by Federal troops would often make it difficult to enforce. They wouldn't clearly win until 1876 and the Compromise of 1877.

After that it has a rather limited history. It provided the Constitutional justification to a Federal law that basically restated the section. It was relevant to Elk v. Wilkins (1884), where the Supreme Court declared Indians could become citizens under the processes there and that they had to count fully. This effectively repealed the 3/5ths Compromise. Saunders v. Wilkins (1945, and a different Wilkins) argued that representation should be redistributed due to poll taxes but it was dismissed by a lower court.

Meanwhile, it has a small cottage industry in feminist criticism. It's the only part of the Constitution that is effectively gendered. In theory, a state could ban women from office without violating the amendment. It was used to justify women's lack of voting rights in 1875 and was part of the reason the Nineteenth Amendment was necessary. Theoretically, it leaves the door open to (for example) banning women from holding elected office since holding office is not among the rights of citizenship. However, no one has ever actually tried that. The Supreme Court has declared that the amendment does not allow discrimination against African Americans and would likely extend that to include women if challenged.

Also, it's been cited numerous times to justify felony disenfranchisement. This is unsurprising since it explicitly allows it. There are only two conditions that currently prevent voting de jure: lack of citizenship and felony disenfranchisement. Additionally, territories do not effectively have a vote. However, as they do not receive apportionment and do not ban people from voting, the Fourteenth Amendment isn't relevant.

It's also received some discussion on the debate of whether illegal immigrants and legal non-citizens should count for apportionment. They currently do but since they cannot vote it's been argued they shouldn't count for apportionment.

From The Right to Vote and Judicial Enforcement of Section Two of the Fourteenth Amendment, Reconstruction, Felon Disenfranchisement, and the Right to Vote, Reconstruction: America's Unfinished Revolution, and Civil Rights Since 1787.