هذه مهمة جديدة.

It is a nominal sentence in Arabic, meaning a sentence without a present-tense verb like is.

Its structure is:

• هذه = this
• مهمة جديدة = a new task

So the sentence literally works like:

• This — a new task

In natural English, that becomes This is a new task.

Because مهمة is a feminine singular noun, and هذه is the feminine singular word for this.

Arabic demonstratives must agree with the gender and number of the thing they refer to.

• هذا = this for masculine singular
• هذه = this for feminine singular

Since مهمة is feminine, Arabic uses هذه.

A big clue is the ending ة (called taa marbuuTa / tāʾ marbūṭa), which often marks feminine nouns.

So:

• مهمة = a feminine noun
• therefore it takes feminine agreement, such as هذه and جديدة

Not every feminine noun ends this way, but many do, and it is one of the most useful signs for learners.

In Arabic, the present-tense verb to be is usually not stated in simple sentences like this.

So Arabic says:

• هذه مهمة جديدة
• literally: This a new task

But the meaning is:

• This is a new task

This is completely normal in Arabic. If you wanted past tense, then a form of كان might appear, for example:

• كانت هذه مهمة جديدة = This was a new task

Because in Arabic, adjectives normally come after the noun they describe.

So:

• مهمة جديدة = a new task

not the English order new task.

This is the normal Arabic pattern:

• noun + adjective

Examples:

• كتاب جديد = a new book
• سيارة جديدة = a new car
• مهمة جديدة = a new task

Because Arabic adjectives must agree with the noun they describe.

Since مهمة is feminine singular, the adjective must also be feminine singular:

• مهمة = feminine singular noun
• جديدة = feminine singular adjective

This is called agreement.

In Arabic, adjectives usually agree in:

• gender
• number
• definiteness
• case (in fully inflected Arabic)

Because here مهمة جديدة means a new task, which is indefinite.

So:

• مهمة جديدة = a new task
• المهمة الجديدة = the new task

An important point: in this sentence, هذه is not directly making the noun phrase this task. Instead, the whole sentence means:

• This is a new task

If you wanted this new task, as one noun phrase, you would say:

• هذه المهمة الجديدة

So there is a difference between:

• هذه مهمة جديدة = This is a new task
• هذه المهمة الجديدة = This new task

مهمة usually means task, mission, or assignment, depending on context.

In many learning contexts, the most natural translation is:

• task
• sometimes assignment
• sometimes mission

So هذه مهمة جديدة could be understood as:

• This is a new task
• This is a new assignment
• This is a new mission

The exact English choice depends on the situation.

In fully vowelled Modern Standard Arabic, it is:

• هٰذِهِ مُهِمَّةٌ جَدِيدَةٌ

Both مهمة and جديدة take ـٌ because they are indefinite and nominative in this sentence.

In normal writing, those short vowels are usually omitted, so you normally see:

• هذه مهمة جديدة

Because this is a nominal sentence, and مهمة جديدة is the predicate of the sentence.

In full case marking:

• هذه = demonstrative pronoun
• مهمةٌ جديدةٌ = nominative predicate

And because جديدة describes مهمة, it matches its case too.

So in full MSA inflection:

• هذه مهمةٌ جديدةٌ

In everyday pronunciation, learners will often hear the sentence without clearly pronounced case endings.

The ending ة (taa marbuuTa) is usually pronounced like -a or -ah when pausing.

So in normal pause, you might hear something like:

• muhimmah
• jadīdah

But when full grammatical endings are pronounced, that ة is pronounced with a t sound before the case ending:

• مهمةٌ = muhimmatun
• جديدةٌ = jadīdatun

So for learners, a simple rule is:

• in pause: often sounds like -a / -ah
• with full endings: behaves like -t

Then the demonstrative and adjective would also change to masculine.

For example:

• هذا كتاب جديد
• This is a new book

Compare:

• هذه مهمة جديدة = feminine
• هذا كتاب جديد = masculine

So Arabic changes more than English does, because agreement is very important.

Yes, in careful Modern Standard Arabic, هذه is commonly pronounced hādhihi.

In slower or fully careful reading, that is the form learners are usually taught.

In pause or in less formal pronunciation, you may hear slight reductions, but hādhihi is the standard form to learn first.