По принцип в този контекст означава "уговаря/настоява/ходатайства" някой да направи нещо, но в случая мисля, че би звучало добре като "предлага екслузивитет".

Трудови, бизнес... - това Вие ще си прецените според контекста.

Nonsolicitation agreement: an agreement by which an employee agrees not to solicit and/or not to accept business from the employer's customers.
https://en.wikipedia.org/wiki/Non-compete_clause

В българското право това като цяло се нарича забрана за конкуренция и не се прави разлика кой кого, но мисля, че израза, който предлагам, ще свърши работа.

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Note added at 1 heure (2018-01-28 20:47:24 GMT)
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Ужас - "изразът" - пълен член :)

"Ковенантът да спазва договора за изключителността на правата (ексклузивноста) или да не предявява искания"

so·lic·it
səˈlisit/Submit
verb
verb: solicit; 3rd person present: solicits; past tense: solicited; past participle: solicited; gerund or present participle: soliciting
ask for or try to obtain (something) from someone.
"he called a meeting to solicit their views"
synonyms: ask for, request, seek, apply for, put in for, call for, press for, beg, plead for More
ask (someone) for something.
"historians and critics are solicited for opinions by the auction houses"
accost someone and offer one's or someone else's services as a prostitute.
"although prostitution was not itself an offense, soliciting was"
Origin
late Middle English: from Old French solliciter, from Latin sollicitare ‘agitate,’ from sollicitus ‘anxious,’ from sollus ‘entire’ + citus (past participle of ciere ‘set in motion’).
Translate solicit to
verb
1. прося
2. моля
3. настоявам
4. ходатайствувам
5. пледирам
6. моля настоятелно
7. защитавам


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Note added at 3 days 23 hrs (2018-02-01 19:10:24 GMT)
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No Solicitation of Customers, Suppliers, Vendors. During the term of this Agreement and for a period of one (1) year after its termination for any reason, Employee agrees not to in any manner, whether directly or indirectly, solicit, contact, or initiate communications with customers, prospective customers, suppliers, or vendors of Company, or request, advise, or entice any such individual to cease doing business with Company, except in the performance of Employees duties under this Agreement.