Is there in english a word connotating the idea ''try to tackle a difficult problem once more''? The word I want has meaning more than simply ''try again''.
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I am not sure about a single word, but some phrases you could use:
Make another attempt.
Work harder to solve ...
Take another stab at it.
Give it another go.
I would like to suggest the phrase "to reengage the problem."
How about persevere:
to continue doing something or trying to do something even though it is difficult (Merriam-Webster Online)
Note that persevere can also be used for continuous successful (but difficult) action, rather than just repeated unsuccessful attempts, but I would say it definitely meets your needs.
I've just heard this expression at BBC Radio 4:
To emphasize the hardness of the task, one may add an adjective before cherry, such as tough, hard.
Let's have a second bite at this tough cherry.
Or, to substitue the word cherry with something a bit tougher. Say, borrow from the idiom 'a hard nut to crack'.
Let's have a second bite at this hard nut, hope it will crack this time.
P.S. Personally I like Obfuskater's reengage: the word has military connotations ("engage the enemy") and thus seems to imply hardness.
In English, we have a prefix to describe a verb as being retried
Retry is the word you are looking for
Other examples of using this prefix: