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To endeavor to locate or obtain someone or something. Often used in passive constructions. The political party has sought after control of parliament for years. We've been seeking after a new HR manager, but none of the candidates have been a good fit so far. The new toy is expected to be heavily sought after this holiday season.
In demand; very popular with or desired by a lot of people. To the surprise of some, the product has been greatly sought after since it was first released. Retailers have struggled to maintain stock of the sought after toy. Now that the property market has begun to boom, construction workers are much sought after once again.
seek after someone or something
to keep looking for someone or something. I will continue to seek after the thief who stole my car. The thief was seeking after a late-model sedan.
Also, much sought after. Very popular, in demand, as in He was much sought after as a throat specialist, particularly by singers. This expression uses the past participle of seek in the sense of "desired" or "searched for." [Late 1800s]