The pioneer asked an old man if he would mind to carry his school bag to the apartment.

The pioneer wanted his school bag carried to the apartment.

The Fly asked the Ant to invite The Dragon-fly to her birthday.

The Fly wanted The Dragon-fly was on to be at the Fly's birthday.

The mother of two warned the judge that she should believe not a single word of her ex-husband.

Я не вижу общего объекта для “The mother of two” и “the judge”.

Mother of two wanted her ex-husband to be distrusted. ← Это получился complex object. (ЮН)

Mother of two wanted her ex-husband distrusted. ← сказуемое тоже нужно заменить. (ЮН)

The mother of two wanted her ex-husband's testimony rejected.

The mother of two wanted to have her ex-husband's words distrusted.

Mr. Black needs it that the radio repare shop should fix his TV-set for him.

Mr. Black needs his TV-set fixed.

The child called and asked to take him home from school.

Я, конечно же сделаю, но это будет комлекс обжект: The child asked to be taken home from school.
Т.е. убрать частицу инфинитива никак не получится. Да и по смыслу получается, что ребенок же мог сам самостоятельно выполнить действие.

The housewife called a plumber to remove the blockage of the tolet basin.

The housewife wanted the blockage of the tolet basin removed.

The child begged his parents to take the street kitten home.

The child begged his parents to take the street kitten home. - В этом случае ребенок тоже может выполнить самостоятельно действие - у него на это есть необходимые знания? навыки и не требуется специальное оборудование.
Другое дело, если фразу подать так: The child begged his parents to take the street kitten to the vet to vaccinate. тогда каузатив хорошо получается: The child wanted to get the street kitten vaccinated.

My son asked me to send his car to the mechanic to change the oil and do the other minor services.

My son want to have his motor oil changed and his car served and the other minor services done.

The defendant asked the court not only to fully justify him but also to pay him compensation for moral harm.

The defendant wanted to have the accusation fully justified and the moral harm compensated.

She asked her stylist to put varnish on her nails of hands and feet.

She wanted all of her nails varnished.

The prosecutor demanded that the developer stop the construction of the house to prevent human casualties.

The prosecutor needed the construction of the house stopped to prevent human casualties.

Mr Black needs it that the vet should vaccinate his pet for him.

Mr Black needs his pet vaccinated.

Mr White tried to convince Mr Black that he should watch the news.

Каузативом не могу сделать, сделаю комплексом.
Mr White wanted Mr Black to watch the news.

She asked him to dye her hair.

She wanted her hair dyed.

Mr.Yellow asked Mr. Brown to paint his submarine.

Mr.Yellow needed his submarine painted.

I asked my tattoo designer that he should make a tattoo on my shoulder.

I wanted my shoulder tattooed.

I begged Father McKenzie absolve me of all my sins.

I wanted all my sins absolved.