How to enlarge a section of y-axis?
I would strongly discourage you from making discontinuous axis, it's going to be very confusing for a reader. 

The facet plot you proposed seems like a good solution to me. Alternatively you can use log transformation. To demonstrate I made it on simulated data that look appox like yours :

set.seed(940401)

data.plot &lt;- data.frame(Methyl_Average = c(0.05, rbeta(699, 2, 5) + 0.15, runif(300, 0.18, 0.85)),
                        Average_Gene_Expression = c(165, exp(rnorm(999, 3))))

library(ggplot2)
require(gridExtra)

g &lt;- ggplot(data.plot, aes(x = Methyl_Average, y = Average_Gene_Expression))
g1 &lt;-  g + geom_point(color = "steelblue") + theme_bw(base_family = "Times") +
       ggtitle( "Scatter Plot") +
       theme_bw(base_family = "Times") +
       theme(plot.title = element_text(hjust = 0.5))

sc_plot &lt;- g1 +
           labs(x = "Differential Methylation", y = "Differential Expression")
logsc_plot &lt;- g1 + 
              labs(x = "Differential Methylation", y = "Differential Expression (log10 scale)") + 
              scale_y_continuous(trans='log10')

grid.arrange(sc_plot, logsc_plot, ncol = 2)