To what extent do gonads affect differences in gendered behaviour?
Yes

During embryonic life,there is formation of new synapses by a process termed as exuberant synaptogenesis.
Their differential formation in the males and females could be due to the effect produced by the gene PCDH11X/Y located in the allosomes.


  PCDH11Y is a gene unique to human males that encodes Protocadherin 11Y, aprotein that guides the development of nerve cells. PCDH11X, located on the X chromosome, is common, in both sexes, to humans and our nearest relative, thechimpanzee; however, PCDH11Y, located on the Y chromosome, is unique to males.[3]
  ...PCDH11X/Y are cadherin family genes. They make proteins, involved in signalling, that attach to the surface of nerve cells.[4] PCDH11X and PCDH11Y, respond in different ways to Retinoic acid, a chemical involved in the development of embryos. The acid stimulates the activity of PCDH11Y but suppresses PCDH11X. This is likely one of the explanations for the differences between the brains of men and women.[3]
  [source]


About the parts of the brain which are deemed"sexually dimorphised" a few of them are:

1.hemispherical differences due to which inter hemispherical functioning is better is females
 whereas intra hemispherical functioning in males

2.hippocampal and amygdaloid laterisation


  on average males had larger grey matter volume in bilateral amygdalae,hippocampi, anterior parahippocampal gyri, posterior cingulate gyri, precuneus,putamen and temporal poles, areas in the left posterior and anterior cingulate gyri
  ...females on average had larger grey matter volume at the rightfrontal pole, inferior and middle frontal gyri, pars triangularis, planum temporale/parietal operculum, anterior cingulate gyrus, insular cortex, and Heschl's gyrus; bilateral thalami andprecuneus; the left parahippocampal gyrus and lateral occipital cortex
  Source:Neuroscience of sex differences


Furthur,since synaptogenesis may occur throughout one's lifespan perhaps not as greatly as during embryonic life and as receptors for gonadal hormones are present in the brain it can be said that the gonads may affect the phenomenon,though mainly after puberty when their concentrations are higher than during childhood.