As others have commented, one approach would be to start by making your datetime data continuous. This will ultimately help with the graph output by creating records for all times of day. When Hang.Seng values do not exist, Hang.Seng will be NA and no data will be displayed (rather than linking these gaps over time with a straight line).

You can do this easily with the (super useful) package padr, which will "pad", or fill out, your time series with the minimum timestep in the starting dataset, giving you a complete, regularly-spaced, continuous time record.

library(tidyverse)
library(lubridate)
library(padr)

hongkongstocks %>% 
  pad() %>%
  ggplot(aes(x=Date, y=Hang.Seng)) + 
  geom_line()+
  scale_x_datetime(limits = c(as_datetime("2020-05-25 00:00:00"), as_datetime("2020-05-26 23:55:00")), 
                   date_breaks = 'day', 
                   date_labels = '%a')

BUT, this graph has gaps in it even during the day, when the market is open. Creating a continuous data set exposed other gaps in the data. If you want to close those gaps in the same way your original graph automatically did (by drawing a straight line between available data points), you can. One option is to create an additional variable that defines when the market is "open" and "closed" (I picked 9:00 - 16:00), and then remove only those records where Hang.Seng is NA, but the market is open. This way ggplot will fill in only the gaps during open hours, but will not connect data points overnight, when the market should be closed.

library(hms)
library(zoo)

hongkongstocks %>% 
  pad() %>%
  mutate(Time = as_hms(Date), #create a separate Time variable
         market_status = if_else((Time >= as_hms("09:00:00") & Time <= as_hms("16:00:00")), "open", "closed")) %>% # create a new market_status variable based on Time
  filter((market_status == "open" & !is.na(Hang.Seng)) | market_status == "closed") %>% # remove records where Hang.Seng is NA, but only when market is open
  ggplot(aes(x=Date, y=Hang.Seng)) + 
  geom_line()+
  scale_x_datetime(limits = c(as_datetime("2020-05-25 00:00:00"), as_datetime("2020-05-26 23:55:00")), 
                   date_breaks = 'day', 
                   date_labels = '%a') +
  labs(x = "Day")

Data

hongkongstocks <- structure(list(Date = structure(c(1590399000, 1590399300, 1590399600, 
1590399900, 1590400200, 1590400500, 1590400800, 1590401100, 1590401400, 
1590402000, 1590402300, 1590402600, 1590403200, 1590403500, 1590403800, 
1590404100, 1590404400, 1590404700, 1590405000, 1590405300, 1590405600, 
1590406200, 1590406800, 1590407100, 1590407400, 1590408000, 1590411900, 
1590412200, 1590412500, 1590412800, 1590413100, 1590413400, 1590413700, 
1590414600, 1590414900, 1590415200, 1590415500, 1590415800, 1590416100, 
1590416700, 1590417000, 1590417300, 1590417600, 1590417900, 1590418200, 
1590418500, 1590418800, 1590419100, 1590419400, 1590419700, 1590421200, 
1590421500, 1590421800, 1590422100, 1590422400, 1590422700, 1590485700, 
1590486000, 1590486300, 1590486600, 1590486900, 1590487200, 1590487500, 
1590487800, 1590488100, 1590488400, 1590488700, 1590489600, 1590490200, 
1590490500, 1590490800, 1590491100, 1590491400, 1590491700, 1590492000, 
1590492900, 1590493200, 1590494100, 1590498300, 1590498600, 1590498900, 
1590499200, 1590499800, 1590500100, 1590500400, 1590500700, 1590501000, 
1590501300, 1590501600, 1590501900, 1590502200, 1590502800, 1590503100, 
1590503400, 1590503700, 1590504000, 1590504300, 1590504600, 1590505200, 
1590506400, 1590507900, 1590508500), tzone = "UTC", class = c("POSIXct", 
"POSIXt")), Hang.Seng = c(100, 98.28, 98.46, 99.11, 99.74, 100.04, 
99.63, 99.77, 99.34, 99.37, 99.06, 99.13, 98.76, 98.72, 98.62, 
98.74, 98.64, 98.71, 98.93, 99.23, 98.99, 99.09, 99.02, 99.05, 
99.04, 99, 99.24, 99.19, 99.27, 99.32, 99.3, 99.33, 99.49, 99.26, 
99.21, 99.35, 99.53, 99.48, 99.51, 99.5, 99.57, 99.61, 99.76, 
99.75, 99.83, 99.97, 100.08, 99.96, 99.88, 99.87, 99.94, 99.98, 
99.99, 100.06, 100.12, 100.1, 101.41, 101.78, 102.05, 101.83, 
101.6, 101.82, 101.77, 101.92, 101.9, 101.98, 101.97, 101.86, 
101.61, 101.79, 101.8, 101.93, 101.99, 101.84, 101.74, 101.85, 
101.88, 101.94, 102.18, 102.09, 102.01, 102.02, 101.95, 101.96, 
102.06, 102.12, 102.1, 102.22, 102.17, 102.26, 102.23, 102.24, 
102.27, 102.3, 102.39, 102.36, 102.34, 102.25, 102.21, 102.13, 
102.04, 102.14)), class = c("spec_tbl_df", "tbl_df", "tbl", "data.frame"
), row.names = c(NA, -102L), spec = structure(list(cols = list(
    Date = structure(list(), class = c("collector_character", 
    "collector")), `Hang Seng` = structure(list(), class = c("collector_double", 
    "collector"))), default = structure(list(), class = c("collector_guess", 
"collector")), skip = 1), class = "col_spec"))