Definition of press on phrasal verb from the Oxford Advanced Learner's Dictionary

 

press on

 phrasal verb
phrasal verb
 
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Phrasal Verbs

press on

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press on with something

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press ahead

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press ahead with something

 
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to continue doing something in a determined way; to hurry forward The company is pressing ahead with its plans for a new warehouse. ‘Shall we stay here for the night?’ ‘No, let's press on.’

press something on somebody

 
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to try to make somebody accept something, especially food or drink, although they may not want it She kept pressing cake on us.
See the Oxford Advanced American Dictionary entry: press on