setainusi to tudukigara wo kakuninsuru.

It is read:

せたいぬし と つづきがら を かくにんする。

A couple of useful notes:

• 世帯主 = せたいぬし
• 続柄 = つづきがら
• 確認する = かくにんする

A learner might expect 主 to be read しゅ, but in 世帯主 it is normally ぬし.

世帯主 means the head of household in an official or administrative sense.

It does not necessarily mean:

• the oldest person
• the person who owns the home
• the person who earns the most money

It means the person registered as the household’s representative for things like government records, residence registration, insurance, and forms.

続柄 means relationship or family relationship, especially in official paperwork.

In this kind of context, it usually means something like:

• spouse
• child
• parent
• sibling

So on a form, 続柄 often means relationship to the head of household.

This word is more formal and administrative than a general word like 関係.

Here, と means and.

It links the two nouns:

• 世帯主 = head of household
• 続柄 = relationship

So 世帯主と続柄 means the head of household and the relationship.

In this sentence, と is not acting as:

• with
• a quotation marker
• a conditional

It is simply joining two nouns in a list.

Because 世帯主と続柄 is treated as one combined object.

So the structure is:

• [世帯主と続柄] を 確認する
• [head of household and relationship]
  • confirm

Japanese often puts the particle only once, after the whole noun phrase, instead of after each item.

This is very natural.

The dictionary form 確認する is often used in:

• instructions
• manuals
• checklists
• menus
• headings
• system messages

So here it can feel like:

• Confirm the head of household and relationship
• Check the head of household and relationship

It does not have to mean a plain casual sentence in conversation.

Other possible versions would be:

• 世帯主と続柄を確認します。 = polite statement
• 世帯主と続柄を確認してください。 = please confirm

Yes, it is grammatically complete.

Japanese often leaves out the subject when it is obvious from context. So this can stand on its own, especially in written instructions.

The implied meaning could be something like:

• Please confirm the head of household and relationship
• You should check the head of household and relationship

In natural Japanese, this kind of omission is very common.

That is a very good question, because the two expressions are similar but not identical.

• 世帯主と続柄を確認する = confirm the head of household and the relationship as two items
• 世帯主との続柄を確認する = confirm the relationship to the head of household

So the version in your sentence sounds like checking two separate fields, which is very common on forms or screens.

If the writer wanted to focus specifically on the relationship to the head of household, 世帯主との続柄 would be more explicit.

It is common in official contexts, but not as common in casual everyday conversation.

You often see 続柄 in:

• government forms
• hospital paperwork
• insurance documents
• resident records

In everyday speech, people are more likely to use expressions like:

• どういう関係ですか = what is your relationship?
• 家族との関係 = relationship to the family member
• 親子関係 = parent-child relationship

So 続柄 sounds a bit formal and administrative.

を marks the direct object of the verb 確認する.

The thing being confirmed is:

世帯主と続柄

So the pattern is:

• X を 確認する
• confirm X

This is a very common structure in Japanese. For example:

• 名前を確認する = confirm the name
• 住所を確認する = confirm the address
• 内容を確認する = confirm the details