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Your LimeSurvey version: Evalution-Question
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Survey theme/template: Evalution-Question
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I would have the following requirements, among others.

I would like to define questions that the end user sees and receives points for answering.

But there should also be text fields where the user can enter any text.

The administrator should then read through these texts afterwards and, depending on the content, also have the option of awarding extra points for them. These points should then be automatically added to the remaining points. It would be best if the administrator could set the points himself or have a choice.

Is this scenario supported at all and if so, what options would I have?

You can create questions and award points, either the old fashioned and a little restrictive way through assessments or by writing your own rules for distributing the points in equation type questions.

You can of course add text fields.

However, there is no built in option for a administrator to review the text answers and give points to it.

You might be able to add a hidden text / number or equation question after each text question and you could try to add points into these hidden questions by editing the responses, but I don't think it is a very straight forward process.