Question

Jenny hopes that Mr Smith will suggest a good way to have her written English improved in a short

词句中谓语动词是哪个啊？
hope/ suggest / have 这些词在句子中分别做什么成分？

Answer 1

Jenny hopes that Mr Smith will suggest a good way to have her written English improved in a short period.
Jenny希望Smith先生会建议一个令她英文写作短期内提高的好办法。

主句很简单，是主谓宾结构：
Jenny(主) hopes(谓) that ...(宾语从句)

that引导的宾语从句又是主谓宾结构：
that Mr Smith(主) will suggest(谓，一般将来时) a good way(宾) to have ...(不定式作定语修饰a good way)

不定式：
to have (不定式非谓语动词) her written English(非谓语动词have的宾语) improved(过去分词非谓语动词作宾语补语补充说明her written English) in a short period(介词短语作improved的状语).

to have something done/improved/fixed/made/... 是一个固定用法。再给你举几个例子：

I'm going to have my hair cut.
John had all his money stolen.
How many authors have had their lives made into movies?

追问

请问， 什么时候用过去分词做宾补 什么时候用现在分词 什么时候用不定式做宾补呢？

追答

这句宾语和宾补的关系是主表关系，即缺了系动词的主系表结构。所以加上系动词就是一个完整的句子：
Her written English is improved.

宾语用现在分词的宾补是主谓的现在进行时：
Her written English is improving.
这个意思和原句后面的时间状语(短时间内)不符。

宾补用不定式的时候，宾语和宾补是主谓关系：
He asked me to come. (I come)
回到你的句子，have something done是固定用法，不能用不定式。